\section{Introduction}
\label{sec:Intro}

The fractional quantum Hall effect (FQHE) at filling factor $\nu = 1/3$ in the lowest Landau level (LLL) was the first to be experimentally observed~\cite{Tsui82} and subsequently understood, as a result of Laughlin's trial wave function~\cite{Laughlin83}. The FQHE has since proven to be an incredibly rich platform for exploring the physics of strongly correlated electron systems. After the initial observation, many additional fractions were observed in the LLL at $\nu = n/(2pn\pm1)$ with $n$ and $p$ positive integers. These fractions are understood as an integer quantum Hall effect (IQHE) of composite fermions, where a composite fermion (CF) is an emergent particle consisting of an electron bound to an even number of vortices~\cite{Jain89, Jain07, Halperin20}.

In contrast, the FQHE in the second LL (SLL) of GaAs is less well understood. Interestingly, even the physical origin of the $\nu=7/3$ FQHE, which corresponds to 1/3 filled SLL, has not been conclusively established. Exact diagonalization studies~\cite{Ambrumenil88, Peterson08, Peterson08b, Balram13b, Kusmierz18, Balram20} have convincingly shown that the actual state at $\nu=7/3$ for a zero width system is an incompressible FQHE state. However, the overlap of the exact ground state with the Laughlin state is not large, typically less than $60\%$ for systems accessible to exact diagonalization studies~\cite{Ambrumenil88, Peterson08, Peterson08b, Balram13b, Kusmierz18, Balram20}. [In contrast, the overlap of the Coulomb ground state at $\nu=1/3$ in the LLL with the Laughlin state is greater than $98\%$ for up to $N=15$ electrons~\cite{Balram20}.] Furthermore, the excitations of the 7/3 FQHE in exact diagonalization studies are qualitatively different from those at 1/3, and exact diagonalization studies also do not show a clearly identifiable branch of low-energy excitations, called the magnetoroton or the CF-exciton mode, as observed in the LLL~\cite{Girvin85, Girvin86, Dev92, Scarola00, Jolicoeur17}. As a result, the precise nature of the state at $\nu=7/3$ has remained a topic of debate~\cite{Read99, Balram13b, Johri14, Zaletel15, Peterson15, Jeong16, Balram19a, Balram20}.

In a recent work, Balram \emph{et al}.\cite{Balram19a} have proposed, inspired by the parton paradigm for the FQHE~\cite{Jain89b}, that the $\nu=7/3$ FQHE is a $\mathbb{Z}_n$ topological superconductor, wherein bound states of $n$ composite bosons~\cite{Zhang89} undergo Bose-Einstein condensation. This generalizes the Zhang-Hansson-Kivelson theory of the 1/3 Laughlin state as a Bose-Einstein condensate of composite bosons~\cite{Zhang89}, with $\mathbb{Z}_1$ corresponding to the Laughlin wave function. While the different $\mathbb{Z}_n$ states share many topological quantum numbers, a key distinction between them is that the elementary quasiparticle has a charge of $-e/(3n)$, where $-e$ is the charge of the electron. Variational calculations in Ref.~\cite{Balram19a} suggest that the best candidate is the $\mathbb{Z}_3$ state, which has lower energy than the Laughlin state in the thermodynamic limit, and also a higher overlap with the exact SLL Coulomb ground state for systems where such a calculation is possible.

The experimental observations at $\nu = 7/3$ are, however, largely consistent with the Laughlin state. In particular, shot noise~\cite{Dolev08, Venkatachalam11,Dolev11} and scanning single-electron transistor~\cite{Venkatachalam11} experiments at $\nu = 7/3$ have measured quasiparticles of charge $-e/3$. This raises the question: Why are experimental measurements consistent with the Laughlin state while theory suggests that better variational states exist? This question has motivated the present study.

There can be several reasons for the discrepancy between theory and experiment. The theoretical calculations mentioned above do not include the effects of finite width, Landau level mixing, screening, and disorder, which can affect the variational comparisons. We consider in this article the competition between the different $\mathbb{Z}_n$ states as a function of the quantum well width.  Our primary result is the prediction of a phase transition from the $\mathbb{Z}_4$ state at small widths into the Laughlin ($\mathbb{Z}_1$) state when the quantum well width exceeds approximately 1.5 magnetic lengths. We also predict a similar phase transition at $\nu=1/3$ in the zeroth Landau level of bilayer graphene as a function of the magnetic field.

\section{$\mathbb{Z}_n$ parton wave function at $\nu=1/3$}
\label{sec:wftns}
The parton theory generalizes the Jain CF states~\cite{Jain07} to a larger class of candidate wave functions~\cite{Jain89b}. In the parton theory, one considers fractionalizing electrons into a set of fictitious particles called partons. The partons are fractionally charged, have the same density as electrons, and have filling factor $\nu_\alpha$, where $\alpha$ labels the parton species.  An incompressible state is achieved when each parton species is in an IQHE state, i.e. $\nu_\alpha = n_\alpha$, with $n_\alpha$ an integer. (More generally, we can place the partons in any known incompressible states.) The partons are of course unphysical and must be combined back into physical electrons, which is equivalent to setting the parton coordinates $z_j^\alpha$ equal to the parent electron coordinates $z_j$, i.e. $z_j^\alpha = z_j$ for all $\alpha$. (The quantity $z_j = x_j - iy_j$ is the complex coordinate of the $j$th electron.) The resulting wave functions, labeled ``$n_1n_2n_3...$," are given by
\begin{equation}
\Psi^{n_1n_2n_3...}_\nu = \mathcal{P}_{\rm LLL} \prod_{n_\alpha} \Phi_{n_\alpha}(\{z_j\}),
\end{equation}
where $\Phi_n$ is the Slater determinant wave function for the state with $n$ filled Landau levels, and $\mathcal{P}_{\rm LLL}$ denotes projection into the LLL, as appropriate in the high field limit. The partons can also experience magnetic fields anti-parallel to the field experienced by electrons; these correspond to negative filling factors, which we denote as $\bar{n}$, with $\Phi_{\bar{n}}=\Phi_{-n}=\Phi_n^*$. To ensure that each parton species has the same density as the electron density, the charge of each parton species is given by $e_\alpha = -\nu e / \nu_\alpha$. The relation $\sum_\alpha e_\alpha=-e$ implies that the electron filling factor is given by $\nu = [\sum_\alpha \nu_\alpha^{-1}]^{-1}$. The Laughlin wave function at $\nu=1/3$ can be interpreted as the $111$ parton state. The Jain $n/(2pn+1)$ states appear as the $n11...$ states and the Jain $n/(2pn-1)$ states as $\bar{n}11...$; these correspond to the wave function $\Psi_{n/(2pn\pm1)} = \mathcal{P}_{\rm LLL} \Phi_{\pm n} \Phi_1^{2p}$. Many other parton states have recently been shown to be plausible for SLL and other FQHE~\cite{Wu17, Balram18, Bandyopadhyay18, Balram18a, Faugno19, Balram19, Kim19, Balram19a, Balram20, Faugno20a, Balram20a}. These states often have exotic properties, such as non-Abelian anyonic excitations~\cite{Wen91}.

For $\nu = 1/3$, Balram \emph{et al}. proposed the $\mathbb{Z}_n$ parton states described by the wave function
\begin{equation}
\Psi_{1/3}^{\mathbb{Z}_n} = \mathcal{P}_{\rm LLL}\Phi_n\Phi_{\bar{n}}\Phi_1^3 \sim \Psi_{n/(2n+1)}\Psi_{n/(2n-1)} \Phi_1^{-1},
\end{equation}
where in the last step we redefine the wave function as $[\mathcal{P}_{\rm LLL}\Phi_n \Phi_1^2] [\mathcal{P}_{\rm LLL}\Phi_{\bar{n}} \Phi_1^2]\Phi_1^{-1}$. (This grouping is chosen to facilitate the use of Jain-Kamilla projection method in our numerical simulations \cite{Jain07,Jain97, Jain97b, Davenport12, Moller05, Balram15a}. It is accepted and has been shown for many cases that the topological properties of the state do not depend on the details of the projection method \cite{Balram16b}.) Because the factor $\Phi_n\Phi_{\bar{n}}$ is real, all the $\mathbb{Z}_n$ states occur at the same ``shift"~\cite{Wen92} $\mathcal{S}=3$ in the spherical geometry. The physical interpretation of the wave function as a superconductor of composite bosons arises from the fact that $\Phi_n\Phi_{\bar{n}}$ represents a $\mathbb{Z}_n$ superconductor of electrons~\cite{Barkeshli13,Balram19a}, and the factor $\Phi_1^3$ attaches three vortices to each electron to convert it into a composite boson. The elementary excitation corresponds to an excitation in the factor $\Phi_n$ or $\Phi_{\bar{n}}$ and has a charge of magnitude $e/(3n)$.

\section{Finite width phase diagram for $\nu=7/3$ in GaAs}

We will use the spherical geometry~\cite{Haldane83} in our calculations, which considers $N$ electrons on the surface of a sphere subjected to a total flux $2Q\phi_0$, with $\phi_0=hc/e$ and $2Q$ is a positive integer. The radius of the sphere is $R=\sqrt{Q}\ell$, where $\ell=\sqrt{\hbar c/eB}$ is the magnetic length. The state with $n$ filled Landau levels can only be constructed for particle number $N$ divisible by $n$ and $N\geq n^2$. The same is true of the $\mathbb{Z}_n$ state.

We approximate the confining potential as an infinite square well of width $w$ that results in a transverse wave function given by a sine function. The problem of electrons in the SLL interacting with the Coulomb interaction is equivalent to that of electrons in the LLL interacting with an effective interaction; the effective interaction for this system is given in Ref.~\cite{Toke08}. We consider well widths up to 5 magnetic lengths. (For convenience, we use the disk pseudopotentials for our calculations in the spherical geometry; this should not cause any corrections because we will perform the calculation for large systems and take the thermodynamic limit.)

\begin{figure}[tbhp]
\includegraphics[width=0.49\textwidth]{SLLTL}
\caption{Thermodynamic extrapolations for the energy differences $\Delta E=E(\mathbb{Z}_n)-E_{\rm Laughlin}$ for various $\mathbb{Z}_n$ states at $\nu=7/3$. The results are for zero quantum well width. The $\mathbb{Z}_4$ state has the lowest energy in the thermodynamic limit.}
\label{fig: TLs}
\end{figure}

\begin{figure}[tbhp]
\includegraphics[width=0.49\textwidth]{SLLFW}
\caption{Thermodynamic energies of various $\mathbb{Z}_n$ states at $\nu=7/3$ as a function of the quantum well width $w$. All energies are quoted relative to the Laughlin state.  The $x$-axis is the quantum well width in units of magnetic length. A phase transition from the $\mathbb{Z}_4$ to the Laughlin state is seen to occur at $w\sim1.5 \ell$.}
\label{fig: FWSLL}
\end{figure}

At zero width, the $\mathbb{Z}_4$ state has the lowest energy in the thermodynamic limit, as seen in Fig.~\ref{fig: TLs}. Here and below, all energies are quoted in units of $e^2/(\epsilon \ell)$ where $\epsilon$ is the dielectric constant of the material. (We note that the $\mathbb{Z}_4$ and $\mathbb{Z}_5$ states were not studied in Ref.~\cite{Balram19a}. Also, we cannot definitively rule out the $\mathbb{Z}_3$ state within the numerical accuracy of our calculations.) We have similarly determined the thermodynamic energies for quantum wells of various widths [see Appendix~\ref{sec: ThermoLim}]. In Fig.~\ref{fig: FWSLL}, we show how the energies of several $\mathbb{Z}_n$ states, measured relative to the energy of the Laughlin state, evolve as we increase the well-width. For a $w<1.5\ell$, the $\mathbb{Z}_4$ state has the lowest variational energy in the thermodynamic limit. For $w>1.5\ell$, the Laughlin state is preferred in our calculation, suggesting that the $\mathbb{Z}_4$ state should only be observed in samples with sufficiently low quantum well widths and/or low density. We add here that because of the numerical uncertainty in the thermodynamic energy differences and our simple model for the finite width, the critical value of $1.5\ell$ should be taken only as a first estimate.

\section{$\mathbb{Z}_n$ state in bilayer graphene}

We next ask if similar physics can appear elsewhere. We expect to find a transition in the zeroth Landau level of bilayer graphene (BLG) as a function of the magnetic field. The zeroth Landau level of BLG is exactly equivalent to the LLL of GaAs when the magnetic field is infinite and continuously interpolates to the SLL of GaAs as the magnetic field is decreased. As such, we expect that a $\mathbb{Z}_n$ state is stabilized below a critical field and the Laughlin state is favored above the critical field.

The Coulomb interaction between electrons can be parameterized using Haldane pseudopotentials $V_m$~\cite{Haldane83}, which is the energy of two electrons in a relative angular momentum state $m$ in the disk geometry. The pseudopotentials in the zeroth LL of bilayer graphene are given by
\begin{equation}
V^{0-{\rm BLG}}_m(\theta)  = \int_{0}^{\infty} dq~F^{0-{\rm BLG}}(\theta,q)e^{-q^2}L_m(q^2).\label{BLGVm}
\end{equation}
where the Fourier transformed form factor is \cite{Apalkov11}
\begin{equation}
F^{0-{\rm BLG}}(\theta,q) = \Bigg[\sin^{2}(\theta)L_{1}\Big(\frac{q^2}{2}\Big)+\cos^{2}(\theta)L_{0}\Big(\frac{q^2}{2}\Big) \Bigg]^2.
\label{eq:ZerothLL_BLG_form_factor}
\end{equation}
Here we have set the magnetic length to unity, $L_r(x)$ is the $r$th order Laguerre polynomial and $\theta$ is a parameter that varies between $0$ and $\pi/2$ to control the relative proportion of the $n=0$ and $n=1$ LLs in the two-component wave function. At $\theta=0$ the form factor is that of the LLL of GaAs while for $\theta = \pi/2$, it is exactly the form factor for the SLL in GaAs. At the mid-way point, $\theta =\pi/4$, the form factor is that of the $n=1$ LL in monolayer graphene~\cite{Balram15c}. The value of $\theta$ is related to the magnetic field by $\tan^2{(\theta)}\propto \ell /(\hbar v_F)\propto 1/\sqrt{B}$, where $v_F$ is the Fermi velocity. For very large magnetic fields, we anticipate the physics of the LLL of GaAs. As the magnetic field is lowered, we first expect to see the physics of monolayer graphene appear (which has been shown to be well described by the composite fermion theory~\cite{Balram15c}) that eventually gives way to states exhibiting the physics of the SLL of GaAs at very small magnetic fields.

\begin{figure}[tbhp]
\includegraphics[width=0.4\textwidth]{BiGrNoSpinEdit}
\caption{Energies of $\mathbb{Z}_n$ states in bilayer graphene, measured relative to the energy of the Laughlin state, as a function of the tangent of the mixing angle $\theta$. All energies represent thermodynamic limits. The $\mathbb{Z}_4$ state is seen to be favored for $\theta \gtrapprox 1.45$. The top axis shows $\hbar v_F/\ell$ in units of meV (see text for relation between $\theta$ and $\hbar v_F/\ell$). Energies are shown only in the vicinity of the transition. }
\label{fig:BiGr}
\end{figure}

We construct an effective interaction as shown in Appendix~\ref{sec: effInt}, and obtain the thermodynamic energies of various candidate states as a function of $\theta$.  The angle $\theta$ is related to measurable quantities through $\tan{\theta}=  t \ell/(\sqrt{2}\hbar v_F)$ where $t$ is the hopping integral and $v_F$ is the Fermi velocity~\cite{Apalkov11}. Taking $t\sim 350\text{meV}$, as obtained from DFT calculations at zero magnetic field~\cite{Jung14}, we obtain $\hbar v_F/\ell = 350\text{meV}/(\sqrt{2}\tan{\theta})$. The top axis in Fig.~\ref{fig:BiGr} shows $\hbar v_F/\ell$ in units of meV.  We find that the transition from the $\mathbb{Z}_{4}$ to the Laughlin state occurs approximately at $\hbar v_F/\ell\sim30$ meV. For graphene, with a typical Fermi velocity of $10^6$ $m/s$, this corresponds to a magnetic field strength of $B\approx 1.4$T.

\section{Landau level mixing and spin}

It is natural to ask if $\mathbb{Z}_n$ parton states can be relevant for $\nu=1/3$ in the LLL. We have performed extensive calculations as a function of quantum well width and density, also including LL mixing. We use a self-consistent LDA calculation to determine the transverse electron density at zero magnetic field at several electron densities and quantum well widths. We further include LL mixing through the so-called fixed phase diffusion Monte Carlo method~\cite{Ortiz93, Zhang16, Zhao18}. For all parameters we have considered, the Laughlin state remains the lowest energy state. The detailed results are given in Appendix~\ref{sec: ThermoLim}.

Recent experiments have mapped out the spin polarization of the SLL in GaAs quantum wells~\cite{Yoo19}. They observe an anomalous spin depolarization between fillings factors 1/5 and 1/3. The Laughlin state is fully spin polarized, but $\mathbb{Z}_n$ states with $n>1$ allow for the possibility of spin-unpolarized or spin-partially polarized states. The generalization is analogous to that for Jain CF states to spin-singlet or partially spin-polarized states~\cite{Wu93, Park98, Jain07, Balram15a, Balram17}. Specifically, the $\mathbb{Z}_n$ state can be generalized to include spin as
\begin{eqnarray}
\Psi_{1/m}^{\mathbb{Z}_n(n,0;\bar{n}_\uparrow,\bar{n}_\downarrow)} &=& \mathcal{P}_{\rm LLL}\Phi_n\Phi_{\bar{n}_\uparrow,\bar{n}_\downarrow}\Phi_1^m \nonumber \\
&\sim & \Psi_{n/(2n+1)} \Phi_1^{m-4} \mathcal{P}_{\rm LLL} \Phi^*_{n_\uparrow,n_\downarrow} \Phi_1^2
\label{Znspin}
\end{eqnarray}
where $n =n_\uparrow + n_\downarrow$, and $\Phi_{n_\uparrow,n_\downarrow}$ represents the state with $n_\uparrow$ spin-up and $n_\downarrow$ spin-down filled Landau levels. These wave functions can be shown to satisfy the Fock cyclic conditions~\cite{Jain07}.
In the above wave function, we have made the $\bar{n}$-parton spinful. An analogous wave function $\Psi_{1/m}^{\mathbb{Z}_n(n_\uparrow,n_\downarrow;\bar{n},0)}$ can be written where the $n$-parton is endowed with spin. Which configuration is preferred depends on the interaction. Our detailed calculations, shown in Appendix~\ref{sec: ThermoLim}, demonstrate that the fully spin-polarized states have better variational energies for all interactions considered in this article.

\section{Discussion}
\label{sec: discussion}
Our work was motivated by an apparent discrepancy between theory and experiment for the FQHE at $\nu=7/3$: while theory finds the $\mathbb{Z}_4$ parton state to have lower energy than the Laughlin state, experiments are consistent with the latter. We find that when we take into account finite width corrections, there is a transition from the $\mathbb{Z}_4$ state into the Laughlin state at width $\sim 1.5 \ell$.

All experimental observations of the 7/3 state appear to be for larger widths and thus fall in the region where the Laughlin state is favored. (Large mobilities, necessary for an observation of the 7/3 state, are typically obtained for relatively wide quantum wells because that minimizes the effect of interface roughening. One may alternatively go to low densities. The 7/3 state has been observed at very low densities~\cite{Pan14, Samkharadze17}, but even there, with 7/3 state occurring at $B\approx 0.9$T, the width of 65 nm translates  approximately into 2.5 $\ell$.) It may be possible to decrease both the quantum well width and the density to get into the regime where the $\mathbb{Z}_4$ state is predicted. If a phase transition is observed (for example, by gap closing and reopening as a function of the density), it would provide evidence in favor of an unconventional $\mathbb{Z}_n$ state at small widths, and also of the role of large width in stabilizing the Laughlin state at 7/3.

We note that a previous exact diagonalization calculation has also shown that the magnetoroton branch, absent at zero width, appears by the time the quantum well width is three magnetic lengths~\cite{Jolicoeur17}. Another exact diagonalization study has shown that finite width stabilizes the 7/3 Laughlin state~\cite{Balram20}.

An additional experimental quantity, namely the chiral central charge, can be measured in thermal Hall conductance measurements and can sometimes distinguish between states with different topological content~\cite{Kane97, Cappelli02}. For the $\mathbb{Z}_n$ states, the chiral central charge is independent of the value of $n$ and so the thermal Hall conductance is predicted to be the same for all of these states. The measured value of the thermal Hall conductance at $7/3$ is consistent with all of these states \cite{Banerjee17b}.

The Hall viscosity of all $\mathbb{Z}_n$ states is identical because they all have the same shift. These states are, however, not topologically equivalent as they have different topological entanglement entropies~\cite{Balram19a}. The clearest experimental signature distinguishing the states will be the charge of the fundamental quasiparticles. The Laughlin state has charge $-e/3$ quasiparticles while the $\mathbb{Z}_n$ state has charge $-e/(3n)$ quasiparticles. These quasiparticles can, in principle, be detected through scanning electron transistor experiments~\cite{Venkatachalam11}. The situation for shot noise experiments is more subtle. As Balram \emph{et al}. argued~\cite{Balram19a}, the $-e/(3n)$ quasiparticles are gapped at the edge and only the $-e/3$ quasiparticles can be excited at arbitrarily low temperatures.  It may be possible, however, that the $-e/(3n)$ quasiparticles become relevant in shot noise experiments at somewhat elevated temperatures (or voltage bias).

We note that the so-called anti-Read-Rezayi $4$-cluster (aRR$4$) state~\cite{Read99} also provides a plausible candidate wave function for the $7/3$ FQHE~\cite{Peterson15}. The energy of the aRR$4$ state is equal to the energy of the Laughlin state within numerical uncertainty~\cite{Peterson15}, in contrast to our $\mathbb{Z}_4$ state which has lower energy than Laughlin's. Furthermore, the aRR$4$ state has overlaps of 0.77 and 0.59 with the exact ground state for 10~\cite{Kusmierz18} and 12 particles, whereas the $\mathbb{Z}_2$ state has an overlap of 0.87 for 10 particles and $\mathbb{Z}_3$ has an overlap of 0.93 for 9 particles~\cite{Balram19a}. (The $\mathbb{Z}_4$ state requires a minimum of 16 particles, for which we cannot obtain overlaps.) Finally, assuming equilibration of all edge modes, the thermal Hall measurements at 7/3 are inconsistent with the chiral central charge of the aRR$4$ state~\cite{Banerjee17b}.

\begin{acknowledgments}
The work at Penn State was supported by the U. S. Department of Energy, Office of Basic Energy Sciences, under Grant no. DE-SC0005042. Some portions of this research were conducted with Advanced CyberInfrastructure computational resources provided by The Institute for CyberScience at The Pennsylvania State University. Some of the numerical calculations reported in this work were carried out on the Nandadevi supercomputer, which is maintained and supported by the Institute of Mathematical Science’s High Performance Computing center. One of us (Th.J.) acknowledges CEA-DRF for providing CPU time on the supercomputer COBALT at GENCI-CCRT.
\end{acknowledgments}


\section{Introduction}

All graphs in this paper will be finite, simple, and undirected.
We use \cite{hhs1} as a reference for terminology and notation which are not explicitly defined here.
For a graph $G = (V(G), E(G))$, let $\pi$ be a graphical property that can
be possessed, or satisfied by the subsets of $V$.
 For example, being a maximal complete subgraph, a maximal independent set,
 acyclic,  a closed/open neighborhood, a minimal dominating set, etc.
Suppose that $f_\pi$   and $F_\pi$ are  the associated  graph invariants:
 the minimum  and maximum cardinalities of a set with property $\pi$.
Let $\mu \in \{f_\pi, F_\pi\}$. For a graph $G$, denote by $\mathtt{M}_\mu(G)$
the family of all  subsets of $V(G)$ each of which has property $\pi$ and cardinality $\mu(G)$.
Each element of $\mathtt{M}_\mu(G)$ is called  a $\mu$-{\em set} of $G$.
Fricke et al. \cite{fhhhl} define a graph $G$ to be $\mu$-{\em excellent} if
each its vertex belongs to some $\mu$-set.
Perhaps historically the first results on $\mu$-excellent graphs
were published by  Berge \cite{berge}
who  introduced the class of $B$-graphs consisting of all
graphs in which every vertex is  in a maximum independent set.
Of course  all $B$-graphs form  the class of $\beta_0$-excellent graphs,
where $\beta_0$ stand for the  independence number.
The study of excellent graphs with respect to the  some domination related parameters
was initiated by   Fricke et al. \cite{fhhhl} and continued e.g. in \cite{bs,hh,h2,msu,sam3,sry,ys}.

 In this paper  we focus on the following  subclass of the class of $\mu$-excellent graphs.

\begin{definition} \label{def1}
Let $\mathcal{I}$ be a set of all mutually nonisomorphic graphs and $\emptyset \not= \mathcal{H} \subsetneq \mathcal{I}$.
We say that a   graph $G$ is   $\mathcal{H}$-$\mu$-excellent  if the following hold:
\begin{itemize}
\item[(i)]  For each $H \in \mathcal{H}$ and for each $x \in V(G)$
									there exists an induced subgraph $H_x$ of $G$
                   such that $H$ and $H_x$ are isomorphic,  $x \in V(H_x)$ and
									$V(H_x)$ is a subset of some  $\mu$-set of $G$.
\item[(ii)] For each  induced subgraph $H$  of $G$,
										which is isomorphic to some element of $\mathcal{H}$,
                     there is  a $\mu$-set of $G$ having $V(H)$ as a subset.
\end{itemize}
\end{definition}

By the above definition it immediately follows that each  $\mathcal{H} $-$\mu$-excellent graph
is $\mu$-excellent. If a graph $G$ is  $\mathcal{H} $-$\mu$-excellent
and $\mathcal{H}$ contains only one element,
e.g.   $\mathcal{H} = \{H\}$, we sometimes omit the brackets
and say that a graph $G$ is $H$-$\mu$-excellent.
Define the $\mu$-{\em excellent family of induced subgraphs} of a $\mu$-excellent graph $G$,
denoted by $G\left\langle \mu \right\rangle$, as the family of all graphs
$H \in \mathcal{I}$ for which $G$ is $H$-$\mu$-excellent.
The next  two observations are  obvious.

\begin{observation}  \label{mu}
If $G$ is a $\mu$-excellent  graph, then $\{K_1\} \subseteq G\left\langle \mu \right\rangle$
 and  $\mu(G)  \geq \max \{|V(H)| \mid H \in G\left\langle \mu \right\rangle\} $.
\end{observation}

\begin{observation}  \label{munu}
Let a graph   $G$ be both $\mu$-excellent  and $\nu$-excellent.
If the set of all  $\mu$-sets and the set of all $\nu$-sets of $G$ coincide,
then $G\left\langle \mu \right\rangle  =  G\left\langle \nu \right\rangle$.
\end{observation}

As first examples of $\mathcal{H}$-$\mu$-excellent graphs let us consider the case $\mu = \beta_0$.
Clearly, any $\beta_0$-excellent graph $G$ is $\{\overline{K_1}, \overline{K_{\beta_0(G)}}\}$-$\beta_0$-excellent.
A graph is $r$-{\em extendable} if every independent set of size $r$
is contained in a maximum independent set (Dean and Zito \cite{dz}).
Clearly,   a graph  is $\{\overline{K_1},\overline{K_2},..,\overline{K_r}\}$-$\beta_0$-excellent
if and only if  it  is $s$-extendable for all $s=1,2,..,r$.
 Plummer \cite{p} define a graph $G$ to be {\em well covered} whenever
$G$ is $k$-extendable for every integer $k$. In other words,
a graph $G$ is well covered if and only if
$G\left\langle \beta_0 \right\rangle = \{\overline{K_1},\overline{K_2},..,\overline{K_{\beta_0(G)}}\}$.

In this paper we concentrate mainly on excellent graphs with respect to the domination number $\gamma$.
We give basic terminologies and notations in the rest of this section.
In Section $2$ we describe the $\gamma$-excellent
 family of induced subgraphs for some well known graphs.
 In Section $3$ we show  that,  under appropriate restrictions,
the generalized lexicographic product of graphs has
the same excellent family of induced subgraphs
with respect to six domination-related parameters.
Section 4   contains results on $\gamma$-excellent regular graphs and trees.
  We conclude in Section 5 with some open problems.

In a graph $G$, for a subset $S \subseteq V (G)$ the {\em subgraph induced} by $S$ is the graph
$\left\langle S \right\rangle$ with vertex set $S$ and
two vertices in $\left\langle S \right\rangle$ are adjacent if and only if they are adjacent in $G$.
The {\em complement} $\overline{G}$ of $G$ is the graph whose
vertex set is $V (G)$ and whose edges are the pairs of nonadjacent vertices of $G$.
We write $K_n$ for the {\em complete graph} of order $n$ and $P_n$ for the  {\em path}
on $n$ vertrices. Let $C_m$ denote the {\em cycle} of length $m$.
 For any vertex $x$ of a graph $G$, $N_G(x)$ denotes the set of all neighbors
of $x$ in $G$, $N_G[x] = N_G(x) \cup \{x\}$  and the degree of $x$ is $deg_G(x) = |N_G(x)|$.
The {\em minimum} and {\em maximum} degrees
 of a graph $G$ are denoted by $\delta(G)$ and $\Delta(G)$, respectively.
For a subset $S \subseteq V (G)$, let $N_G[S] = \cup_{v \in S}N_G[v]$.
Let $X \subseteq V(G)$ and $x \in X$.
The $X$-{\em private neighborhood} of $x$, denoted by $pn_G[x,X]$ or simply by $pn[x,X]$
if the graph is clear from the context, is the set $\{y \in V(G) \mid N[y] \cap X = \{x\}\}$.
A {\em leaf} is a vertex of degree one and a {\em support vertex} is a vertex adjacent to a leaf.
The $1$-{\em corona}, denoted $cor(U)$, of a graph $U$ is the graph obtained from $U$
 by adding a degree-one neighbor to every vertex of $U$.
An {\em isomorphism} of graphs $G$ and $H$ is a bijection between the vertex sets of $G$ and $H$
$f \colon V(G)\to V(H)$ such that any two vertices $u$ and $v$ of $G$
 are adjacent in $G$ if and only if $f(u)$ and $f(v)$ are adjacent in $H$.
If an isomorphism exists between two graphs, then the graphs are called isomorphic
 and denoted as $ G\simeq H$. We use the notation $[k]$  for $\{1,2,..,k\}$.

An  {\em independent set}  is a set of vertices in a graph, no two of which are adjacent.
The {\em independence number} of $G$, denoted $\beta_0(G)$, is the maximum size of an independent set in $G$.
The {\em independent domination number} of $G$, denoted by $i(G)$, is the minimum size of a maximal independent set in $G$.
A subset $D \subseteq V(G)$ is called a {\em dominating set} (or a {\em total dominating set}) in $G$, if
for each $x \in  V (G) -D$ (or for each $x \in V (G)$, respectively) there exists a vertex $y \in D$ adjacent to $x$.
A  dominating set $R$ of a graph $G$  is a {\em restrained dominating set} (or an {\em outer-connected dominating set}) in $G$,    if
every vertex  in $V(G)-R$ is adjacent to a vertex in $V(G)-R$ (or $V(G)-R$ induces a connected graph, respectively).
The minimum number of vertices of a dominating set in a graph $G$ is  the {\em domination number} $\gamma(G)$ of $G$.
 Analogously the {\em total domination number} $\gamma_t(G)$,  the {\em restrained domination number} $\gamma_r(G)$  and
the {\em outer-connected domination number} $\gamma^{oc}(G)$ are defined.
The minimum cardinality of a set $S$ which is simultaneously total dominating and restrained dominating  in $G$
 is called  the  {\em total restrained domination number} $\gamma_{tr}(G)$ of $G$.
The minimum cardinality of a set $S$ which is simultaneously total dominating and outer-connected dominating  in $G$
 is called  the  {\em total outer-connected domination number} $\gamma_{t}^{oc}(G)$ of $G$.

\section{\sc Examples}

Here we find the $\gamma$-excellent family of induced subgraphs of some well known graphs.

\begin{example}\label{e12}
Let $G$ be a connected graph with $\gamma(G)=2$.
In \cite{jay} it is proved that (in our terminology)  $G$ is $K_2$-$\gamma$-excellent if and only if
$G$ is a complete $r$-partite graph $K_{n_1,n_2,..,n_r}$, $n_i \geq 2$, $i=1,2,..,r \geq 2$.
Clearly $K_{2,2,..,2}\left\langle \gamma \right\rangle = \{K_1,K_2, \overline{K_2}\}$ and
$K_{n_1,n_2,..,n_r}\left\langle \gamma \right\rangle = \{K_1,K_2\}$ when $n_s \geq 3$ for some $s \in [r]$.
\end{example}

\begin{example}\label{pncr}
Let  $\nu \in \{\gamma, i\}$. Then all the following hold:
\begin{itemize}
\item[(i)] (folklore)  $\nu(P_n) = \left\lceil n/3 \right\rceil$ and $\nu(C_r) = \left\lceil r/3 \right\rceil$.
                                    $C_r$ is $\nu$-excelent for all $r \geq 3$.
                                       $P_n$ is $\nu$-excellent if and only if $n=2$ or $n \equiv 1 \pmod 3$.
\item[(ii)]  $P_n\left\langle \nu \right\rangle = \{K_1\}$ when $n \in \{1,2\} \cup \{7,10,\dots\}$
                     and $P_4\left\langle \nu \right\rangle = \{K_1,\overline{K_2}\}$
\item[(iii)] $C_5\left\langle \nu \right\rangle = \{K_1, \overline{K_2}\}$ and
                     $C_{3r}\left\langle \nu \right\rangle =  C_{5+3r}\left\langle \nu \right\rangle = \{K_1\}$, $r \geq 1$.
\item[(iv)]  $C_7\left\langle \gamma \right\rangle = \{K_1, K_2,\overline{K_2},\overline{K_3}\}$,
                     and $C_{3r+1}\left\langle \gamma \right\rangle =\{K_1, K_2, \overline{K_2}\}$ for $r \not=2$.
\item[(v)]  $C_7\left\langle i \right\rangle = \{K_1, \overline{K_2},\overline{K_3}\}$
                     and $C_{3r+1}\left\langle i \right\rangle =\{K_1,  \overline{K_2}\}$  for $r \not=2$.
\end{itemize}
\end{example}

The proof is straightforward and hence we omit it.
From the above example, one can easily obtain the next result.

\begin{example}\label{disc1}
Let a graph $G$ be an union of $s \geq 2$ paired disjoint cycles $C_{n_1},C_{n_2},..,C_{n_s}$.
\begin{itemize}
\item[(i)] If $n_i=5$, $i=1,2,..,s$, then $G\left\langle \gamma \right\rangle =  \{\overline{K_1},\overline{K_2},..,\overline{K_{2s}}\}$.
\item[(ii)] If $n_i=7$, $i=1,2,..,s$, then $G\left\langle \gamma \right\rangle= \{\overline{K_1},\overline{K_2},..,\overline{K_{3s}}\} \cup \{K_2\}$.
\item[(iii)] If $n_i \not\equiv 1 \pmod 3$ and $n_i \geq 6$ for some $i \in [s]$, then  $G\left\langle \gamma \right\rangle =\{K_1\}$.
\item[(iv)] If $n_i \equiv 1 \pmod 3$  and $n_i \geq 10$ for all $i \in [s]$, then
                    $G\left\langle \gamma \right\rangle = \{K_1, K_2, \overline{K_2}\}$.
\end{itemize}
\end{example}

Denote by (CEA) the class of all graphs $G$ such that $\gamma(G+e) \not= \gamma(G)$ for all $e \in E(\overline{G})$.
\begin{example}\label{e2}
Let a noncomplete graph $G$ be in (CEA).
 It is well known fact that any two nonadjacent vertices of $G$ belong to some $\gamma$-set of $G$ (Sumner and Blitch \cite{sb}).
In other words,  $G$ is $\{K_1, \overline{K_2}\}$-$\gamma$-excellent graph.
\end{example}

\begin{proposition}\label{g=b0}
Let $G$ be a graph with $\beta_0(G) = \gamma(G)=s$.
Then $G$ is $\{\overline{K_1}, \overline{K_2},..,\overline{K_s}\}$-$\gamma$-excellent
 and \cite{p} $G\left\langle i \right\rangle = G\left\langle \beta_0  \right\rangle= \{\overline{K_1},..,\overline{K_s}\}$.
\end{proposition}
\begin{proof}
Every independent set of $G$ is a subset of a maximal independent set.
Since each maximal independent set is always dominating and $\beta_0(G) = \gamma(G)=s$,
the result immediately follows.
\end{proof}

The {\em Cartesian product} of two graphs $G$ and $H$ is the graph $G\square H$ whose
vertex set is the Cartesian product of the sets $V (G)$ and $V (H)$. Two vertices
$(u_1, v_1)$ and $(u_2, v_2)$ are adjacent in $G \square H$ precisely when either
 $u_1 = u_2$ and $v_1v_2 \in  E(H)$ or $v_1 = v_2$ and $u_1u_2 \in E(G)$.
 It is clear from this  definition that $G\square H \simeq H\square G$ and
if $G$ or $H$ is not connected then $G \square H$ is not connected.

\begin{example}\label{Knn}
Let $G = K_m\square K_n$, $n\geq m \geq 2$.
Then  $G\left\langle i \right\rangle = G\left\langle \beta_0 \right\rangle= \{\overline{K_1},..,\overline{K_m}\}$.
If $n > m$, then $G\left\langle \gamma \right\rangle = \{\overline{K_1},..,\overline{K_m}\}$.
If $n=m$, then
$G\left\langle \gamma \right\rangle = \{\overline{K_1},..,\overline{K_m}\} \cup \{K_1,K_2,..,K_m\}
\cup \{K_p \cup \overline{K_q} \mid (p \geq 2) \wedge (q \geq 1) \wedge (p+q \leq m)\}$.
\end{example}

 \begin{proof}
Let $G = K_m\square K_n$, $n\geq m \geq 2$.
 We consider $G$ as an $m \times n$ array of vertices
$\{x_{i,j} \mid (1 \leq i \leq m)  \wedge (1 \leq j \leq n)\}$,
 where the closed neighborhood of $x_{i,j}$ is the union of
the sets  $A_i = \{x_{i,1}, x_{i,2},..,x_{i,n}\}$ and $B_j = \{ x_{1,j}, x_{2,j},..,x_{m,j}\}$.
Then $\left\langle A_i \right\rangle \simeq K_m$ and $\left\langle B_j \right\rangle \simeq K_n$.
It is well-known that  \cite{gr1} (a) $\gamma(G) = i(G) = \beta_0(G) = m$,
(b) $A_1,A_2,..,A_m$ are $\gamma$-sets of $G$,
and if $m=n$, $B_1,B_2,..,B_n$ are also  $\gamma$-sets of $G$.
Hence, by Proposition \ref{g=b0}, $G$ is $\{\overline{K_1}, \overline{K_2},..,\overline{K_m}\}$-$\gamma$-excellent
and $G\left\langle i \right\rangle = G\left\langle \beta_0 \right\rangle=  \{\overline{K_1},..,\overline{K_m}\}$.
Suppose that $G$ is $H$-$\gamma$-excellent.
Then there is a $\gamma$-set $D$   of $G$ such that  $\left\langle D \right\rangle$
has an induced subgraph $H_1 \simeq H$.  Assume that $H$ has at least one edge.
 \medskip

{\em Case} 1: $m < n$.
 Clearly $|A_i \cap  D|=1$  for all $i=1,2,..,m$.
Because of symmetry, we assume without loss of generality that
$D \cap B_j $ is empty for all $j>m$. Define now the set
$D^t = \{x_{r,s} \mid x_{s,r} \in D\}$.  Since  $H$ is not edgeless,
 $|D \cap B_j| > 1$ for some $j \leq m$. But  then $|D^t \cap A_j|>1$,
which means that $D^t$ is not a $\gamma$-set of $G$.
Since $\left\langle D \right\rangle  \simeq \left\langle D^t \right\rangle$,
 $G$ is not $H$-$\gamma$-excellent.
Thus, $G\left\langle \gamma \right\rangle = \{\overline{K_1},..,\overline{K_s}\}$.
\medskip

{\em Case} 2: $m = n$.
Obviously in this case  exactly one of  $|A_i \cap  D|=1$  for all $i=1,2,..,m$
and $|B_j \cap  D|=1$  for all $j=1,2,..,m$ holds.  Say the first is valid.
Let $R_1$ be a   $l$-order component   of $\left\langle H \right\rangle$ for some $l \geq 2$.
For the sake of symmetry, we can  assume  that all  elements of $R_1$ are in $B_1$
 and  $D \subset \cup_{s=1}^pB_s$, where $D \cap B_s$ is not empty for all $s \in [p]$.
Clearly  $p \leq m-l+1$.
Suppose that $\left\langle D \right\rangle$  has another nontrivial component.
Then the difference $m-p$ is not less  than $l$.
Define the set $D_1 = (D - V(R_1)) \cup \{x_{1,p+1},x_{1,p+2},..,x_{1,p+l}\}$.
Clearly $D_1$ is not a $\gamma$-set of $G$ and
$\left\langle D_1 \right\rangle \simeq \left\langle D \right\rangle$.
Thus $R_1$ is the only nontrivial component of $\left\langle D \right\rangle$.
Hence $H$ is either a complete graph or a union of complete and edgeless graph.
Finally, it is easy to see that for each such a graph $H$,
$G$ is $H$-$\gamma$-excellent.
\end{proof}

We  need the following  "negative result".

\begin{theorem}\label{neg}
There is no $P_3$-$\gamma$-excellent graph $G$ with $\gamma(G) = 3$.
\end{theorem}

\begin{proof}
 Assume that $G$ is a $P_3$-$\gamma$-excellent graph, $\gamma(G) = 3$
 and $x_1,x_2,x_3$ is an induced path in $G$.
     Since $X = \{x_1,x_2,x_3\}$ is a $\gamma$-set of $G$,
		there  is $y_i \in pn[x_i, X]$, $i=1,2,3$.
	Then $\{x_1,x_2,y_2\}$ is a $\gamma$-set of $G$, which implies $y_2y_3 \in E(G)$.
		But now  no vertex of the induced path $y_2,y_3,x_3$  is adjacent to $x_1$, a contradiction.

	\end{proof}

\begin{example}\label{compl}
  $\overline{K_3 \square K_n} \left\langle \gamma \right\rangle =
	\{K_1,K_2,\overline{K_2}, K_1 \cup K_2, \overline{K_3}, K_3\}$ when $n \geq 3$,
and   $\overline{K_m \square K_n} \left\langle \gamma \right\rangle =
\{K_1,K_2,\overline{K_2}, K_1 \cup K_2, K_3\}$ when $n \geq m \geq 4$.
\end{example}
\begin{proof}
First note that $\overline{K_3 \square K_3} \simeq K_3 \square K_3$ and by
Example \ref{Knn} it immediately follows that $\overline{K_3 \square K_3} \left\langle \gamma \right\rangle =
	\{K_1,K_2,\overline{K_2}, K_1 \cup K_2, \overline{K_3}, K_3\}$.
	So, let $n \geq 4$ and $n \geq m \geq 3$.
	It is well known that \cite{gr1} $\gamma(\overline{K_m \square K_n}) = 3 \leq m = i(\overline{K_m \square K_n})$.
		Let  us  consider  the graph $G_{m,n} = \overline{K_m \square K_n}$
as  a $m \times n$ array  of  vertices $\{a_{i,j} \mid (1 \leq i \leq m )  \wedge  (1 \leq j \leq n)\}$,
 with  an  adjacency $N(a_{i,j}) = V(G_{m,n} ) - (Y_i \cup Z_j)$,
where  $Y_i = \cup_{k=1}^{n} \{a_{i,k}\}$ and $Z_j = \cup_{r=1}^{m}\{a_{r,j}\}$.
Remark now that:
\begin{itemize}
\item[(a)]    $\left\langle \{a_{i,j}, a_{k,l}, a_{r,s}\} \right\rangle \simeq K_3$
                       if and only if  both $3$-tuples $(i,k,r)$ and $(j,l,s)$ consist of paired distinct integers.
											 The vertices of each triangle of $G_{m,n}$ form a $\gamma$-set.
											Every two adjacent vertices  $a_{i,j}$ and $a_{k,l}$ belong to a triangle.
\item[(b)]    All induced subgraphs isomorphic to $K_1 \cup K_2$ are
                       $\left\langle \{a_{i,j}, a_{k,l}, a_{i,l}\} \right\rangle$ and $\left\langle \{a_{i,j}, a_{k,l}, a_{k,j}\} \right\rangle$,
											where $i \not=k$ and $j \not= l$. The vertices of each such a subgraph form a $\gamma$-set.
											Every two vertices belong to an induced subgraph isomorphic to $K_1 \cup K_2$.
	\item[(c)]  Each $3$-cardinality subset of $Z_j$ is independent and it is not dominating.
		\end{itemize}

Theorem \ref{neg} together with (a)-(c) immediately lead to the required.
\end{proof}

To continue we need the following theorem and definitions.

\begin{them}\label{lbcp} \cite{ep}
 $\gamma (G \square H) \geq \min \{|V(G)|, |V(H)|\}$ for any two  arbitrary graphs $G$ and $H$.
\end{them}

A $G$-{\em layer} of the Cartesian product $G \square H$ is the set
 $\{(u,y)\mid u \in V(G)\}$,where $y \in V(H)$.
Analogously an $H$-{\em layer} is the set
$\{(x,v) \mid v\in V(H)\}$,
where $x\in V(G)$. A subgraph of $G \square H$ induced by a $G$-layer or
 an $H$-layer is isomorphic to $G$ or $H$, respectively.

\begin{theorem}\label{existance}
Let $H$ be a connected noncomplete $n$-order graph and $p \geq n \geq 3$.
If each induced  subgraph of $K_p \square H$ which is  isomorphic to $H$ has as a vertex set some
 $H$-layer, then  $\gamma(K_p \square H)  = n$ and  $K_p \square H$ is a $H$-$\gamma$-excellent graph.
\end{theorem}
\begin{proof}
Each $H$-layer of  $K_p \square H$ is a dominating set  of  $K_n \square H$.
Hence  $\gamma (K_p \square H) \leq |V(H)|=n$. Since $p \geq n$,  by Theorem \ref{lbcp}
we  have that each $H$-layer is a $\gamma$-set of  $K_p \square H$.
It remains to note that clearly each vertex of $K_p \square H$ belongs to some $H$-layer.
\end{proof}

The  next  example serves as an illustration of the above theorem.

\begin{example}\label{kpcn}
If $p \geq n \geq 5$, then the graph  $K_p \square C_n$ is  $C_n$-$\gamma$-excellent.
\end{example}
\begin{proof}
Let $H$ be an induced subgraph of $K_p \square C_n$ which is isomorphic to $C_r$.
It is easy to see that if $H$ is not a $C_n$-layer,   then either $r \in \{3,4\}$ or $r \geq n+2$.
The required immediately follows by Theorem \ref{existance}.
\end{proof}

\section{Generalized lexicographic product}

Let $G$ be a  graph with vertex set
 $V(G) = \{\textbf{1}, \textbf{2},..,\textbf{n}\}$ and   let
 $\Phi = (F_1,F_2,..,F_n)$ be  an ordered $n$-tuple of  paired disjoint graphs.
Denote by $G[\Phi]$ the graph with  vertex set $\cup_{i=1}^n V(F_i)$
and edge set defined as follows: (a) $F_1, F_2,.., F_n$ are induced subgraphs of $G[\Phi]$,
 and  (b) if $x \in V(F_i)$, $y \in V(F_j)$, $i,j \in [n]$ and $i \not= j$, then
$xy \in E(G[\Phi])$ if and only if \textbf{ij} $\in E(G)$.
A graph  $G[\Phi]$ is called  the  {\em generalized lexicographic product}  of $G$ and $\Phi$.
If  $F_i \simeq F$ for every $i=1,2,..,n$, then  $G[\Phi]$
 becomes the standard lexicographic product  $G[F]$.
Each subset $U = \{u_1,u_2,..,u_n\} \subseteq V(G[\Phi])$ such that
                         $u_i \in V(F_i)$, for every $i \in [n]$, is called a $G$-layer.
From the definition of $G[\Phi]$ it immediately follow:
\begin{itemize}
\item[(A)] (folklore) $G[\Phi] \simeq G$  if and only if  $G[\Phi] = G[K_1]$.
                         $G[F] \simeq F$  if and only if $G \simeq  K_1$.
                        If $G$ has at least  two vertices, then $G[\Phi]$ is connected if and only if $G$ is connected.
												If $G$ is edgeless, then $G[\Phi] = \cup_{i=1}^nF_i$.
                        For any $G$-layer $U = \{u_1,u_2,..,u_n\}$
                        the bijection  $f \colon V(G)\to U$ defined by  $f(\textbf{i}) = u_i \in V(F_i)$ is an
										    isomorphism between $G$ and $\left\langle  U \right\rangle$.
												For any $x \in V(F_i)$ and $y \in V(F_j)$, $i \not= j$, is fulfilled
												$dist_{G[\Phi]}(x,y) = dist_G(\textbf{i},\textbf{j})$.
\end{itemize}
 The equality $dist_{G[\Phi]}(x,y) = dist_G(\textbf{i},\textbf{j})$
will be used in the sequel without specific references.

\begin{theorem}\label{three}
Given a graph $G[\Phi]$, where $G$ is  connected of order $n \geq 2$  and
$|V(F_k)| \geq 3$ for all $k \in [n]$.  Then
$G[\Phi]\left\langle \gamma \right\rangle = G[\Phi]\left\langle \gamma_r \right\rangle = G[\Phi]\left\langle \gamma^{oc} \right\rangle$ and
$G[\Phi]\left\langle \gamma_t \right\rangle = G[\Phi]\left\langle \gamma_{tr} \right\rangle = G[\Phi]\left\langle \gamma_t^{oc} \right\rangle$.
If $\gamma(F_k) \geq 3$ for all $k \in [n]$, then
$G[\Phi]\left\langle \gamma \right\rangle = G[\Phi]\left\langle \gamma_r \right\rangle = G[\Phi]\left\langle \gamma^{oc} \right\rangle =
G[\Phi]\left\langle \gamma_t \right\rangle = G[\Phi]\left\langle \gamma_{tr} \right\rangle = G[\Phi]\left\langle \gamma_t^{oc} \right\rangle$.
\end{theorem}
\begin{proof}
Let $\mu \in \{\gamma, \gamma_t\}$  and $D$ a $\mu$-set of $G[\Phi]$.
Assume there is $i \in [n]$ such that $V(F_i) \cap D = \{v_1,v_2,..,v_r\}$, where $r \geq 2$.
Then clearly for each $\textbf{j} \in N(\textbf{i})$, $V(F_j) \cap D$ is empty and for
any $u_j \in V(F_j)$ the set $(D-\{v_2,..,v_r\}) \cup \{u_j\}$ is a dominating set of $G[\Phi]$
or a total dominating set of $G[\Phi]$ depending on whether $\mu = \gamma$ or
$\mu = \gamma_t$, respectively.  Hence $r=2$.  Since $G$ is connected of order $n \geq 2$ and
$|V(F_i)| \geq 3$ for all $i \in [n]$, the graph $\left\langle V(G[\Phi]) - D \right\rangle$ is connected.
Therefore the first two equality chains are correct.

Finally, let $D_1$ be a $\gamma$-set of $G[\Phi]$ and  $\gamma(F_k) \geq 3$ for all $k \in [n]$.
Then clearly for every $i \in [n]$ the sets $D$ and $V(F_i)$  must have no more than one element in common.
But this immediately implies that $D_1$ is a total dominating set of $G[\Phi]$.
Thus, the last equality chain holds.
\end{proof}

\begin{theorem}\label{complete}
Given a graph $G[\Phi]$, where $G$ is  connected of order $n \geq 2$  and  $F_k$ is complete
with $|V(F_k)| \geq 2$ for all $k \in [n]$.  Then $G[\Phi]$ is $\overline{K_s}$-$\gamma$-excellent
if and only if $G$ is $\overline{K_s}$-$\gamma$-excellent.
\end{theorem}
\begin{proof} Recall that any $G$-layer of $G[\Phi]$ induces a graph isomorphic to $G$.  We need the following claim.
\medskip
\\
{\bf Claim 1.}  (i) Each  $\gamma$-set $D$ of $G[\Phi]$ is contained in a $G$-layer of $G[\Phi]$;
                           moreover, $D$ is a $\gamma$-set of each subgraph of $G[\Phi]$ that is induced by
													a $G$-layer  containing $D$.
														(ii) If $D^*$ is a $\gamma$-set of some subgraph of $G[\Phi]$ that is induced by
													      a $G$-layer,  then $D^*$ is a $\gamma$-set of $G[\Phi]$.
\noindent
\begin{proof}[Proof of Claim 1.]
If $D$ is a  $\gamma$-set of $G[\Phi]$, then since all $F_i$'s are complete $|D \cap V(F_i)| \leq 1$ for all $i \in [n]$.
But then $D$ is a dominating set of any subgraph of $G[\Phi]$ that is induced by a $G$-layer containing $D$.
In particular this leads to $\gamma(G[\Phi]) \leq \gamma(G)$.

If $D^*$  is a $\gamma$-set of some subgraph of $G[\Phi]$ that is induced by  a $G$-layer,
then again by the fact that all $F_i$'s are complete, it follows that $D^*$  is a dominating set of $G[\Phi]$.
This clearly leads to  $\gamma(G[\Phi]) \geq \gamma(G)$.

Thus $\gamma(G[\Phi]) = \gamma(G)$ implying the required.
\end{proof}

$\Leftarrow$ Choose $u \in V(G[\Phi])$ arbitrarily. Then there is a $G$-layer $U$ containing $u$.
                           Since $G$ is $\overline{K_s}$-$\gamma$-excellent, there is a  $\gamma$-set $D^*$
													of $\left\langle U \right\rangle$ that contains $s$ paired nonadjacent vertices
													one of which is $u$. By  Claim 1, $D^*$ is a 			$\gamma$-set of $G[\Phi]$.

													If $R$ is a $s$-vertex independent set  in $G[\Phi]$, then since all $F_i$'s are complete graphs,
													$R$ is a subset of some $G$-layer.  The rest is as above.

$\Rightarrow$ Let $L = \{l_1,l_2,..,l_n\}$ be a $G$-layer of $G[\Phi]$, where $l_i \in V(F_i)$, $i \in [n]$.
                              Choose $l_r \in L$ arbitrarily.
                             Since $G[\Phi]$ is $\overline{K_s}$-$\gamma$-excellent, there is an $s$-vertex independent set $I_s$
														of $G[\Phi]$ and a $\gamma$-set $D$ of $G[\Phi]$ such that $u \in I_s \subseteq D$.
														By Claim 1, $D$ is a $\gamma$-set of some subgraph induced by a $G$-layer of $G[\Phi]$.

														 Since all $F_i$'s are complete, without loss of generality, we can assume that $D \subseteq L$.

														 Let $R$ be a $s$-vertex independent set of $L$. Then there is a $\gamma$-set $D_1$ of $G[\Phi]$ which
														has $R$ as a subset. By Claim 1 $D_1$ is a $\gamma$-set of a graph induced by some
                            $G$-layer and as above we can assume that $D_1 \subseteq L$.
\end{proof}

\section{Regular graphs and trees}
To present the next results on regular graphs, we need the following theorem.

\begin{them}\label{degree}
Let $G$ be a $n$-order graph and minimum degree $\delta$. Then
$\gamma(G) \leq n\delta / (3\delta  -1)$ when $\delta \in \{3,4,5\}$
(see \cite{reed}, \cite{sy} and  \cite{xsc}, respectively).
\end{them}

For any $5$-regular graph $G$ with $\gamma(G) = 3$,
the bound stated in Theorem \ref{degree} can be improved by $3$.

\begin{proposition}\label{3-5-10}
Let $G$ be a $5$-regular graph with $\gamma(G) = 3$.  Then $n \geq 12$.
\end{proposition}
\begin{proof}
By Theorem \ref{degree} we have  $n \geq 9$.
  Since  there is no $5$-regular graphs of odd order,  $n \geq 10$ is even.
	Note that there are exactly sixty $5$-regular graphs of order $10$ \cite{merGT, mer}.
 Their  adjacency lists can be found in  \cite{mer}.
A simple verification shows that each of these graphs has  the domination number equals to $2$.
\end{proof}

\begin{figure}[htbp]
	\centering
		\includegraphics{reg9.eps}
	\caption{The two $4$-regular $K_3$-$\gamma$-excellent  graphs of order $9$. The graph  on the right is $K_3 \square K_3$. }
	\label{fig: reg9}
\end{figure}

\begin{theorem}\label{reg11}
Let $G$ be a $s$-regular $K_r$-$\gamma$-excellent $n$-order  connected graph
with $\gamma(G)=r$, where $n > s \geq r \geq 3$.
Then the following assertions hold.
\begin{itemize}
\item[(i)]  $n \leq r(s-r+2)$.
\item[(ii)] If $r=3$, then $s \geq 4$ with equality if and only if $n=9$ and
                  $G$ is one of the graphs depicted in Fig.\ref{fig: reg9}.
\item[(iii)]  If $r=3$ and  $s=5$, then  $n = 12$.
 \end{itemize}
\end{theorem}

\begin{proof}
(i)  Let  $H \simeq K_r$ be a subgraph of $G$. Each vertex of $H$ is adjacent to
$s-r+1$ vertices outside $V(H)$. Hence  $n \leq r + r(s-r+1) = r(s-r+2)$.

(ii)  Since $r=3$, we have $\gamma(G) =3$ and $n \leq 3s-3$.
     By Theorem \ref{degree} we obtain $8 \leq n$ when $s=3$
			and  $9 \leq n$ when $s \geq 4$.
			Thus $s \geq 4$ and if the equality holds, then $n=9$.
			There are exactly $16$ $4$-regular graphs of order $9$ \cite{mer}.
      An immediate verification shows that among them  only  the graphs depicted in Fig.\ref{fig: reg9}
		  are $K_3$-$\gamma$-excellent.

(iii) 			By (i), $n \leq 12$ and by Proposition \ref{3-5-10} , $n \geq 12$.
 \end{proof}

Note that  the connected $5$-regular $K_3$-$\gamma$-excellent graph  depicted in Fig.  \ref{fig:12} has  order $12$.

\begin{figure}[htbp]
	\centering
		\includegraphics{12.eps}
	\caption{a $5$-regular  $K_3$-$\gamma$-excellent connected graph on $12$ vertices}
	\label{fig:12}
\end{figure}

Now we concentrate on graphs having cut-vertices.

Let $G_1, G_2,..,G_k$ be pairwise disjoint connected graphs of order
at least $2$ and $v_i \in V(G_i)$, $i=1,2,..,k$. Then the   {\em coalescence}
 $(G_1 {\cdot}  G_2 {\cdot}...{\cdot} G_k) (v_1, v_2,.., v_k:v)$
 of $G_1,G_2,...,G_k$  via $v_1, v_2,..,v_k$,
is the graph obtained from the union of $G_1,G_2,..,G_k$   by identifying $v_1, v_2,.., v_k$
in a vertex labeled $v$.
If for graphs $G_1, G_2,..,G_k$ is fulfilled $V(G_i) \cap V(G_j) = \{x\}$ when $i,j = 1,2,..,k$ and $i \not= j$,
then the {\em coalescence}  $(G_1 {\cdot}  G_2 {\cdot}...{\cdot} G_k) (x)$
 of $G_1,G_2,..,G_k$ via $x$ is the union of $G_1, G_2,..,G_k$.

Define  $V^-(G)  = \{x\in V(G) \mid \gamma(G-x) < \gamma(G)\}$ and
  $V^=(G)  = \{x\in V(G) \mid \gamma(G-x) = \gamma(G)\}$. It is well known that
$V^-(G)  = \{x\in V(G) \mid \gamma(G-x) +1 = \gamma(G)\}$.
To continue we need the following result:

\begin{lema}\label{exc2} \cite{bcd}
Let  $G = (F \cdot H)(x)$.  Then   $x \in V^-(G)$ if and only if $x \in V^-(F) \cap  V^-(G)$.
Furthermore, if $x \in V^-(G)$, then $\gamma(G) = \gamma(F) + \gamma(H) -1$.
\end{lema}

\begin{theorem}\label{dwe}
Let $G = (G_1 {\cdot}  G_2 {\cdot}...{\cdot} G_k) (x)$,   $x \in V^-(G)$ and
$G_i$ is $H$-$\gamma$-exellent, $i=1,2,..,k$, where $H$ is connected and has no cut-vertex.
Then $G$ is also $H$-$\gamma$-excellent.
\end{theorem}
\begin{proof}
Using induction on $k$ we easily obtain from  Lemma \ref{exc2} that  $\{x\} =  V^-(G_1)  \cap V^-(G_2) \cap ...\cap V^-(G_k)$ and
$\gamma(G) =  \gamma (G_1) + \gamma (G_2) +..+ \gamma (G_k) - k + 1$.
Consider any induced subgraph $R$ of $G$, which  is isomorphic to $H$.
Since $H$ is connected and without  cut-vertices, $R$ is an induced subgraph of some $G_i$,
say without loss of generality, $i=1$.  Then there is a $\gamma$-set $D_1$ of $G_1$
for which $R$ is an induced subgraph of  $\left\langle D_1 \right\rangle$.
Let $D_i$ be a $\gamma$-set of $G_i - x$, $i=2,3,..,k$.
Since  $x \in V^-(G_i)$,  $|D_i| = \gamma(G_j) - 1$. Then
$D = \cup_{i=1}^kD_i$   is a $\gamma$-set of $G$ and $R$ is an induced subgraph of $\left\langle D \right\rangle$.
\end{proof}

Define a {\em vertex labeling}  of a tree $T$  as a function $S:V (T ) \rightarrow\{0,1\}$.
A labeled tree $T$ is denoted by a pair $(T , S)$.
 Let $\textbf{0}_T$ and $\textbf{1}_T$ be the sets of vertices assigned the values $0$ and  $1$, respectively.
	In a {\em labeled} $1$-{\em corona tree}  	of order at least  four all  its leaves
	are in $\textbf{0}_G$ and all  its support vertices form $\textbf{1}_G$.

Let $\mathscr{T}$ be the family of labeled trees $(T, S)$ that can be obtained from a sequence of labeled trees
$\tau: (T^1, S^1), \dots, (T^j, S^j)$, ($j \geq 1$),
such that  $(T^1, S^1)$ is a  labeled   $1$-corona tree of order at least  four
and $(T, S) = (T^j, S^j)$, and, if $j \geq 2$,  $(T^{i+1}, S^{i+1})$ can be obtained recursively from $(T^i, S^i)$
 by  the following operation:

{\bf Operation}   {$O$}.   The labeled tree $(T^{i+1},S^{i+1})$ is obtained from
vertex disjoint  $(T^i,S^i)$ and a labeled     $1$-corona tree $G_i$
in such a way that  $T^{i+1}= (T^i \cdot G_i)(u,v:u)$,
where (a) $u \in \textbf{0}_{T^i}$,  $v \in \textbf{0}_{G_i}$ and $u \in \textbf{0}_{T^{i+1}}$,
and (b) $\textbf{0}_{T^{i+1}} = \textbf{0}_{T^i} \cup \textbf{0}_{G_i}-\{v\}$
 and $\textbf{1}_{T^{i+1}} = \textbf{1}_{T^i} \cup \textbf{1}_{G_i}$.

 Now we are in a position to present a (reformulated)
constructive characterization of $\gamma$-excellent trees.

\begin{them}\label{exc}\cite{sam2}
For any tree $T$ of order at least four the following are equivalent:
\begin{itemize}
\item[(i)] $T$ is $\gamma$-excellent.
\item[(ii)] There is labeling $S:V (T ) \rightarrow\{0,1\}$ such that
                    $(T , S)$ is in $\mathscr{T}$.
\end{itemize}
Moreover, if $(T , S)$ is in $\mathscr{T}$, then $\textbf{0}_{T}= V^-(T)$, $\textbf{0}_{T}$  is a $\gamma$-set of $T$
and $\textbf{1}_{T}= V^=(T)$.  In particular, all leaves of $T$ are in $V^-(T)$.
\end{them}

Another constructive characterization of the $\gamma$-excellent trees  can be found in \cite{bs}.
To prove our last result we need the following lemma.

\begin{lemma}\label{bridge}
Let  $G$ be a connected graph and  $x \in V^-(G)$.
\begin{itemize}
\item[(i)] If $xy$ is a bridge in $G$, then   no $\gamma$-set of $G$ contains both $x$ and $y$.
\item[(ii)]If $xy$ and $xz$ are bridges in $G$, then no  $\gamma$-set of $G$ contains both $y$ and $z$.
 \end{itemize}
\end{lemma}
\begin{proof}
(i)  Clearly, we can consider $G$ as a coalescence $(F\cdot H)(x)$,
       where without loss of generality, $y \in V(F)$ and $x$ is a leaf of $F$.
       Suppose $D$ is a $\gamma$-set of $G$ and $x,y \in D$.
       Then $D \cap V(H)$ and $D \cap V(F)$ are dominating sets of $H$ and $F$, respectively.
        Moreover, since $x$ is a leaf in $F$, $D \cap V(F)$ is not a $\gamma$-set of $F$.
       Hence $|D| = |D \cap V(H)| + |D \cap V(F)| - 1 \geq \gamma(H) + (\gamma(F) +1) -1 $,
        a contradiction with Lemma \ref{exc2}.

(ii) Let as in (i), $G=(F\cdot H)(x)$, $y \in V(F)$ and $x$ is a leaf of $F$.
      Hence $z \in V(H)$.
       Let  	$D$ be a $\gamma$-set of $G$ and $y,z \in D$.
			By (i), $x \not\in D$ and then $D \cap V(H)$ and $D \cap V(F)$ are dominating sets of $H$ and $F$, respectively.
		  This implies $|D| = |D \cap V(H)| + |D \cap V(F)| \geq \gamma(H) + \gamma(F)$,
        a contradiction with Lemma \ref{exc2}.
\end{proof}

\begin{theorem}
 Let $T$ be a $\gamma$-excellent tree of order at least four.
\begin{itemize}
\item[(a)] If $T$ has a cut-vertex belonging to $V^-(T)$, then   $T\left\langle \gamma \right\rangle = \{K_1\}$.
\item[(b)] If no cut-vertex of $T$ is in $V^-(T)$, then  $T$ is a $1$-corona tree and
                    $T\left\langle \gamma \right\rangle = \{\overline{K_1},..,\overline{K_r}\}$, where $2r = |V(T)|$.
\end{itemize}
 \end{theorem}
 \begin{proof}
Suppose $T$ is $H$-$\gamma$-excellent where $H$ is not edgeless.
 Let $D$ be a $\gamma$-set of $T$ and $R \simeq H$ be an induced subgraph
of $\left\langle D \right\rangle$. Choose arbitrarily an edge  $xy$  of $R$.
Clearly both $x$ and $y$ are not leaves and by Lemma \ref{bridge},
neither $x$ nor $y$  is a cut-vertex belonging to $V^-(T)$.
Hence $x,y \in V^=(T)$, because of Theorem \ref{exc}.
Now we choose $xy$ so that $x$ is a leaf in $R$.
By Theorem \ref{exc}, a vertex $y$ has a neighbor $z \in V^-(T)$.
 Lemma \ref{bridge}  now implies  $N[z] \cap D = \{y\}$.
But then  the graph $R_x = \left\langle V(R-x) \cup \{z\} \right\rangle$
is isomorphic to $R$.   Since  $z \in V^-(T)$ and $yz \in E(T)$,
 Lemma \ref{bridge} shows that no $\gamma$-set of $T$ contains
both $y$ and $z$.Thus, we arrive to a contradiction.

Therefore, $T\left\langle \gamma \right\rangle$ contains only edgeless graphs.
By Theorem \ref{exc} $V^-(T)$ is a $\gamma$-set of $T$.
Assume first that there is a cut-vertex $x \in V^-(T)$.
Then for any two neighbors $y$ and $z$ of $x$ the set
$V_1 = (V^-(T) - \{x\}) \cup \{y,z\}$ is independent of cardinality $\gamma(T) + 1$.
 Suppose $T$ is $\overline{K_r}$-$\gamma$-excellent for some $r \geq 2$.
 Choose any cardinality $r$ subset $V_1$ of  $ (V^-(T) - \{x\}) \cup \{y,z\}$
that contains both $y$ and $z$.  Now by Lemma \ref{bridge},
we conclude that no $\gamma$-set of $T$ has $V_1$ as a subset.
Thus, $T\left\langle \gamma \right\rangle = \{K_1\}$.

Finally, let $V^-(T)$ contains only leaves. By Theorem \ref{exc},
$T$ is a $1$-corona tree.  Clearly  $\gamma(T) = i(T) = \beta_0(T) = r$
 and then the required now follows by Proposition \ref{g=b0}.
 \end{proof}

\section{Open problems and questions}
We conclude the paper by listing some interesting problems and directions for further research.

\begin{itemize}
\item[$\bullet$]  For which ordered pairs $(r, s)$ there are
                                 $s$-regular $K_r$-excellent graphs of order $r(s-r+2)$ (see Theorem \ref{reg11})?
                                 Find all  $12$-order $5$-regular  $K_3$-$\gamma$-excellent graphs.
	\end{itemize}

\begin{itemize}
\item[$\bullet$]  Characterize/describe all graphs $F$ such that
                                 there is no $F$-$\mu$-excellent 	graph $G$ with $\mu(G) = |V(F)|$ (see Observation \ref{mu}).
                                  Recall that there is no $P_3$-$\gamma$-excellent graph $G$ with $\gamma(G) = 3$ (Theorem \ref{neg}).
  \end{itemize}

\begin{itemize}
\item[$\bullet$] Let $b$ be a positive integer.
Denote by $\mathscr{A}(\mu,b)$ the class of all $\mu$-excellent connected graphs $G$
for which $\mu(G) = b$ and  $|G\left\langle \mu \right\rangle|$ is maximum.
It might be interesting for the reader to investigate these classes  at least  when $b$ is small.
Note that we already know that
 $\mathscr{ A}(\gamma,1)$ consists of all complete graphs,
and  all connected graphs obtained from  $K_{2n}$, $n \geq 2$,
      by removing a perfect matching form $\mathscr{ A}(\gamma,2)$
      (Example \ref{e12}).
In addition,  by Example \ref{Knn} we	have $\gamma(K_3\square K_3) = 3$,
$K_3\square K_3 \left\langle \gamma \right\rangle =
\{K_1, \overline{K_2}, K_2, K_1 \cup K_2, \overline{K_3}, K_3\}$
and by Theorem \ref{neg} we know that there is no
 $P_3$-$\gamma$-excellent graph $G$ with $\gamma(G) = 3$.
Thus, $K_3 \square K_3$ belongs to $\mathscr{ A}(\gamma,3)$
 and  $|K_3\square K_3 \left\langle \gamma \right\rangle|  = 6$.
Find $\mathscr{ A}(\gamma,3)$.
\end{itemize}

\begin{itemize}
\item[$\bullet$]   Find $T\left\langle \mu \right\rangle$ for each $\mu$-excellent tree  $T$,
                                  where  $\mu \in \{i,\gamma_t, \gamma_R\}$ and $\gamma_R$
																	stand for the Roman domination number
																	(see \cite{hh}, \cite{h2} and \cite{sam3}, respectively).
  \end{itemize}

\begin{itemize}
\item[$\bullet$] Find  graphs $H$ such that  each induced  subgraph of $K_p \square H$
                                 which is  isomorphic to $H$ has as a vertex set some  $H$-layer
																(see  Theorem \ref{existance}).
  \end{itemize}

\begin{itemize}
\item[$\bullet$]  Characterize/describe all connected $\overline{K_2}$-$\gamma$-excellent graphs $G$ with $\gamma(G)=2$.
 \end{itemize}


\section{Introduction}

The LHC will probe the mechanism of electroweak symmetry breaking. A
very attractive alternative to the standard Higgs mechanism, with
fundamental scalars, involves new strongly-interacting gauge theories,
known as technicolor \cite{Weinberg:1979bn, Susskind:1978ms}. Such models
avoid difficulties of theories with scalars, such as triviality and
fine-tuning. Chiral symmetry must be spontaneously broken in a technicolor
theory, to provide the technipions which generate the $W^{\pm}$ and
$Z$ masses and break electroweak symmetry. Although this duplication
of QCD is appealing, precise electroweak measurements have made it
difficult to find a viable candidate theory. It is also necessary to
enlarge the theory (extended technicolor) to generate quark masses,
without generating large flavor-changing neutral currents, which is
challenging.

Technicolor theories have lately enjoyed a resurgence, due to the
exploration of various techniquark representations
\cite{Dietrich:2006cm}. Feasible candidates have fewer new flavors,
reducing tension with electroweak constraints. If a theory is almost
conformal, it is possible this generates additional energy scales,
which could help in building the extended technicolor sector. There
are estimates of which theories are conformal for various
representations, shown in Fig.~\ref{fig:window}. For $SU(N)$ gauge
theory, if the number of techniquark flavors is less than some
critical number, conformal and chiral symmetries are broken and the
theory is QCD-like. For future model-building, it is crucial to go
beyond these estimates and determine precisely where the conformal windows
are. There have been a number of recent lattice simulations of
technicolor theories, attempting to locate the conformal windows for
various representations \cite{Catterall:2007yx, Appelquist:2007hu,
  Shamir:2008pb, Deuzeman:2008sc, DelDebbio:2008zf}.

\begin{figure}[t]
\begin{center}
\includegraphics[width=.4\textwidth]{dietrich_sannino.eps}
\end{center}
\caption{The conformal window for $SU(N)$ gauge theories with $N_f$
  techniquarks in various representations, from
  \cite{Dietrich:2006cm}. The shaded regions are the windows, for
  fundamental (gray), 2-index antisymmetric (blue), 2-index symmetric
  (red) and adjoint (green) representations.}
\label{fig:window}
\end{figure}

\section{Dirac eigenvalues and chiral symmetry}

\begin{figure}[t]
\begin{center}
\includegraphics*[width=.7\textwidth]{quartets_RMT_nX2_b3.9.eps}
\end{center}
\caption{The integrated distribution of the two lowest eigenvalue
  quartets, from simulations of $n_s=2$ Asqtad staggered flavors. This
  is compared to RMT with $N_f=2$ and 8, corresponding to the strong
  and weak coupling limits.}
\label{fig:2stag}
\end{figure}

The connection between the eigenvalues $\lambda$ of the Dirac operator and
chiral symmetry breaking is succinctly given in the Banks-Casher
relation \cite{Banks:1979yr},
\begin{equation}
\Sigma = - \langle \bar{\Psi} \Psi \rangle = \lim_{\lambda \rightarrow
  0} \lim_{m \rightarrow 0} \lim_{V \rightarrow \infty} \frac{\pi
  \rho(\lambda)}{V}.
\label{eq:Banks}
\end{equation}
To generate a non-zero density $\rho(0)$, the smallest eigenvalues
must become densely packed as the volume increases, with an
eigenvalue spacing $\Delta \lambda \approx 1/\rho(0) = \pi/(\Sigma
V)$. This allows a crude estimate of the quark condensate
$\Sigma$. One can do much better by exploring the $\epsilon$-regime:
If chiral symmetry is spontaneously broken, tune the volume and
quark mass such that
\begin{equation}
\frac{1}{F_\pi} \ll L \ll \frac{1}{m_\pi},
\label{eq:epsilon}
\end{equation}
so that the pion is much lighter than the physical value, and
finite-volume effects are dominant \cite{Gasser:1987ah}. The chiral
Lagrangian,
\begin{equation}
{\cal L} = \frac{F_\pi^2}{4} {\rm Tr}( \partial_\mu U \partial_\mu
U^\dagger ) + \frac{\Sigma}{2} {\rm Tr} [ M(U + U^\dagger) ],
\hspace{0.5cm} U = \exp \left[ \frac{i \pi^a T^a}{F_\pi} \right]
\label{eq:chiral}
\end{equation}
is dominated by the zero-momentum mode from the mass term and all
kinetic terms are suppressed. In this limit, the distributions of the
lowest eigenvalues are identical to those of random matrix theory
(RMT), a theory of large matrices obeying certain symmetries
\cite{Shuryak:1992pi}. To connect with RMT, the eigenvalues and quark
mass are rescaled as $z = \lambda \Sigma V$ and $\mu = m \Sigma V$,
and the eigenvalue distributions also depend on the topological charge $\nu$
and the number of quark flavors $N_f$. RMT is a very useful tool to
calculate analytically all of the eigenvalue distributions. The
eigenvalue distributions in various topological sectors are measured
via lattice simulations, and via comparison with RMT, the value of the
condensate $\Sigma$ can be extracted. This method has been
successfully used in a number of lattice QCD studies, for example in
dynamical overlap fermion simulations \cite{Fukaya:2007yv}.

\section{Simulations and analysis}

For $SU(3)$ gauge theory with quarks in the fundamental representation,
various methods suggest that the critical number of flavors
separating conformal and QCD-like behavior is between 8 and 12. In
order to study this interesting region, we simulate $n_s=2$ and 3
staggered fermion flavors, corresponding to $N_f=8$ and 12 flavors in the
continuum limit. (We do not take roots of the determinant of the
staggered Dirac operator). We have also simulated $SU(3)$ gauge theory
with $N_f=2$ flavors in the 2-index symmetric representation, using
dynamical overlap fermions, which is described in \cite{Fodor:2S}. We
use the Asqtad staggered action \cite{Orginos:1999cr}, which includes
improvements to reduce the violations of flavor symmetry (``taste
breaking'') at finite lattice spacing. This action is very well tested
and has been heavily used in large scale simulations of lattice QCD
\cite{Davies:2003ik}. There have been detailed comparisons of
staggered eigenvalues with the Asqtad action to RMT
\cite{Follana:2005km}, but only in the quenched approximation.

\begin{figure}[t]
\begin{center}
\includegraphics*[width=.7\textwidth]{quartets_RMT_nX3_b1.9.eps}
\end{center}
\caption{The integrated distribution of the two lowest eigenvalue
  quartets, from simulations of $n_s=3$ Asqtad staggered flavors. This
  is compared to RMT with $N_f=3$ and 12, corresponding to the strong
  and weak coupling limits.}
\label{fig:3stag}
\end{figure}

Because $n_s=2$ and 3 staggered flavors have not been simulated with
this action before, a large scan of the parameter space of the bare
couplings was required. Hence our first runs were on small volumes
$10^4$, where we also gained experience on the dependence of the
Hybrid Monte Carlo algorithm \cite{Gottlieb:1987mq} on the quark mass
and the discretization of the trajectory length. Once we generated
large thermalized ensembles, we calculated the lowest eigenvalues of the
Dirac operator using the PRIMME package \cite{primme}. In the
continuum limit, the staggered eigenvalues form degenerate quartets,
with restored flavor symmetry. In Figs.~\ref{fig:2stag} and
\ref{fig:3stag}, we show the integrated distributions of the two
lowest eigenvalue quartet averages,
\begin{equation}
\int_0^{\lambda} p_k(\lambda') d\lambda', \hspace{0.5cm} k=1,2
\end{equation}
for ensembles with $n_s=2$ and 3 staggered flavors respectively. Both
simulations have quark mass $ma=0.01$, and the respective bare
couplings are $\beta=3.9$ and 1.9. All low eigenvalues have
small chirality, with no indication of non-zero topology. We see that
the quark mass is less than the average smallest eigenvalue, which is
necessary to probe the behavior of the eigenvalue distributions in the
chiral limit. To compare with RMT, we vary $\mu=m \Sigma V$ until we
satisfy
\begin{equation}
\frac{\langle \lambda_1 \rangle_{\rm sim}}{m} = \frac{\langle z
  \rangle_{\rm rmt}}{\mu},
\label{eq:rmt}
\end{equation}
where $\langle \lambda_1 \rangle_{\rm sim}$ is the lowest quartet average
from simulations and the RMT average $\langle z \rangle_{\rm rmt}$
depends implicitly on $\mu$ and $N_f$. With this optimal value of
$\mu$, we can predict the distributions $p_k(\lambda')$ and compare to
the simulations.

\begin{figure}[t]
\begin{center}
\includegraphics*[width=.7\textwidth]{compare_degeneracy_asqtad_nx1_l10_b6.8.eps}
\end{center}
\caption{Comparison of different improvements of the staggered Dirac
  operator. The eigenvalues are calculated on the same ensemble of
  gauge configurations, which were generated using the Asqtad action.}
\label{fig:impstag}
\end{figure}

In both cases, we see quite good agreement between simulations and RMT
with the corresponding number of flavors in the continuum limit
i.e.~$N_f=8$ and 12. This is somewhat surprising. From the eigenvalues
themselves, one can directly see that flavor breaking is significant,
since degenerate quartets are not yet formed. A previous
eigenvalue study used unimproved staggered quarks in dynamical fermion
simulations \cite{Damgaard:2000qt}. They found excellent agreement
with RMT but only if $N_f$ had the same value as the number of
staggered flavors $n_s$. We also find that, at strong coupling, RMT
with the continuum value $N_f=4 n_s$ does not describe the data. On coarse
lattices, the flavor breaking is very large and only one pion can be
tuned to the $\epsilon$-regime for each staggered flavor. One has to
go to weak coupling and finer lattices, where flavor breaking
decreases, to recover the correct number of light pions.

\begin{figure}[t]
\begin{center}
\includegraphics*[width=.7\textwidth]{stout_degeneracy_stout6x0.15_nx1_l12.eps}
\end{center}
\caption{The lowest eigenvalues calculated on two ensembles with
  $n_s=1$ staggered flavor, with stout smearing used both in the sea
  and valence quark. The lattice volume is $12^4$.}
\label{fig:stout}
\end{figure}

These results indicate that both the $N_f=8$ and 12 flavor
theories with fundamental quarks have a non-zero quark condensate
$\Sigma$ i.e.~chiral symmetry is spontaneously broken.
If this conclusion holds against further studies of flavor breaking
effects, our $N_f = 8$ result will lend considerable support to the
findings of \cite{Appelquist:2007hu, Deuzeman:2008sc}, but the $N_f = 12$
spectrum would be inconsistent with the statement of
\cite{Appelquist:2007hu} that this theory is conformal.

\section{Staggered improvement}

Since flavor breaking can have a dramatic effect on the eigenvalues, we
are investigating various improvements of the staggered action, to
bring the simulations closer to the continuum limit. In
Fig.~\ref{fig:impstag}, we compare mixed actions, with gauge
configurations generated using the Asqtad action, while the
eigenvalues are those of various improved staggered Dirac
operators. This figure is for $n_s=1$ staggered flavor at $\beta=6.8$
and volume $10^4$. The appearance of eigenvalue quartets which are
clearly separated is a clear indication of reduced flavor
breaking. Both HYP-smearing \cite{Hasenfratz:2001hp} and
stout-smearing \cite{Morningstar:2003gk} seem to bring significant
improvement relative to the Asqtad operator, while HISQ fermions
\cite{Follana:2006rc} do not show as clear an improvement.

We also show in Fig.~\ref{fig:stout} the effect of using
stout-smearing both in the sea and valence quark. As we go to weaker
coupling towards the continuum limit, the eigenvalue quartet structure
emerges clearly. Comparison of the improved eigenvalues with RMT is
ongoing.

\section{Conclusions}

Knowledge of the conformal window is essential to build viable
candidates of strongly interacting physics beyond the Standard
Model, and lattice simulations will play a crucial role. Our technique of
studying the eigenvalue properties complements other lattice
approaches, such as calculating the beta function of the renormalized
coupling, looking for finite-temperature transitions, or extracting
the mass spectrum. This will hopefully lead to consensus about the
nature of these new theories. Our first study gives an indication that
$SU(3)$ gauge theory with $N_f=8$ and 12 flavors are both QCD-like,
non-conformal theories. We are investigating various improvements to
reduce flavor-breaking lattice artifacts and allow us to reach a stronger
conclusion.

\section{Acknowledgments}
We thank Poul Damgaard for very helpful discussions, and Urs Heller
who stressed the importance of reaching the quartet degeneracy
limit. This research was supported by the DOE under grants
DOE-FG03-97ER40546, DE-FG02-97ER25308, by the NSF under grant 0704171,
by the DFG under grant FO 502/1 and by SFB-TR/55.