6040880
Rupture degree of graphs
We introduce a new graph parameter, the rupture degree. The rupture degree for a complete graph K n is defined as 1−n, and the rupture degree for an incomplete connected graph G is defined by r(G)=max{ω(G−X)−|X|−m(G−X):X⊂V(G), ω(G−X)>1}, where ω(G−X) is the number of components of G−X and m(G−X) is the order of a largest component of G−X. It is shown that this parameter can be used to measure the vulnerability of networks. Rupture degrees of several specific classes of graphs are determined. Formulas for the rupture degree of join graphs and some bounds of the rupture degree are given. We also obtain some Nordhaus–Gaddum type results for the rupture degree.
1204298
An implicit degree condition for long cycles in 2-connected graphs
Abstract Let id ( v ) denote the implicit degree of a vertex v . In this work we prove that: If G is a 2-connected graph with max { id ( u ) , id ( v ) } ≥ c / 2 for each pair of nonadjacent vertices u and v that are vertices of an induced claw or an induced modified claw of G , then G contains either a Hamilton cycle or a cycle of length at least c . This extends several previous results on the existence of long cycles in graphs.
6391408
Computing the Scattering Number of Graphs
The scattering number of a noncomplete connected graph G is defined by s ( G )= max{ y ( G m X ) m X X ² V ( G ), y ( G m X ) S 2}, where y ( G m X ) denotes the number of components of G m X . This parameter can be used to measure the vulnerability of networks. It shows not only the difficulty to break down the network but also the damage that has been caused. In this paper, we prove that the problem of computing the scattering number of a graph is NP-complete. At the same time, the scattering numbers of grids and those of Cartesian products of two complete graphs are determined.
25468199
Edge vulnerability parameters of split graphs
Abstract A graph is called a split graph if its vertex set can be partitioned into a clique and an independent set. In this work, we investigate three vulnerability parameters of split graphs when edges are removed, i.e., edge-connectivity, edge-toughness and edge-integrity. It is proved that, for a noncomplete connected split graph G , its edge-connectivity is δ ( G ) , and its edge-toughness is min { δ ( G ) , | E ( G ) | | V ( G ) | − 1 } , where δ ( G ) , V ( G ) and E ( G ) , are the minimum degree, the vertex set and the edge set of G , respectively. Furthermore, we show that the edge-integrity of a noncomplete connected split graph equals its order when its minimum degree is greater than half of the size of its largest clique.
39027321
A Fan Type Condition For Heavy Cycles in Weighted Graphs
Abstract. A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree dw(v) of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted graph which satisfies the following conditions: 1. max{dw(x),dw(y)∣d(x,y)=2}≥c/2; 2. w(xz)=w(yz) for every vertex z∈N(x)∩N(y) with d(x,y)=2; 3. In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. Then G contains either a Hamilton cycle or a cycle of weight at least c. This generalizes a theorem of Fan on the existence of long cycles in unweighted graphs to weighted graphs. We also show we cannot omit Condition 2 or 3 in the above result.
1154359
Scattering number in graphs
The scattering number of a noncomplete connected graph G is defined by s(G) = max{ω(G − X) − |X|: X ⊂ V(G), ω(G − X) ≥ 2}, where ω(G − X) denotes the number of components of the graph G − X. In this paper, we show that this parameter can be used to measure the vulnerability of a graph. To some extent, it represents a trade-off between the amount of work done to damage the network and how badly the network is damaged. The relationship between the scattering number and some other parameters of a graph is discussed. Furthermore, we give the Nordhaus–Gaddum-type result for scattering number. © 2001 John Wiley & Sons, Inc.
31017610
Structural Alignment of Pseudoknotted RNA
In this paper, we address the problem of discovering novel non-coding RNA (ncRNA) using primary sequence, and secondary structure conservation, focusing on ncRNA families with pseudo-knotted structures. Our main technical result is an efficient algorithm for computing an optimum structural alignment of an RNA sequence against a genomic substring. This algorithm finds two applications. First, by scanning a genome, we can identify novel (homologous) pseudoknotted ncRNA, and second, we can infer the secondary structure of the target aligned sequence. We test an implementation of our algorithm (Pal), and show that it has near-perfect behavior for predicting the structure of many known pseudoknots. Additionally, it can detect the true homologs with high sensitivity and specificity in controlled tests. We also use Pal to search entire viral genome and mouse genome for novel homologs of some viral, and eukaryotic pseudoknots respectively. In each case, we have found strong support for novel homologs.
3022359
CVSSearch: searching through source code using CVS comments
CVSSearch is a tool that searches for fragments of source code by using CVS comments. CVS is a version control system that is widely used in the open source community. Our search tool takes advantage of the fact that a CVS comment typically describes the lines of code involved in the commit and this description will typically hold for many future versions. In other words, CVSSearch allows one to better search the most recent version of the code by looking at previous versions to better understand the current version. In this paper we describe our algorithm for mapping CVS comments to the corresponding source code, present a search tool based on this technique, and discuss preliminary feedback.
30170890
A new algorithm for generalized fractional programs
A new dual problem for convex generalized fractional programs with no duality gap is presented and it is shown how this dual problem can be efficiently solved using a parametric approach. The resulting algorithm can be seen as “dual” to the Dinkelbach-type algorithm for generalized fractional programs since it approximates the optimal objective value of the dual (primal) problem from below. Convergence results for this algorithm are derived and an easy condition to achieve superlinear convergence is also established. Moreover, under some additional assumptions the algorithm also recovers at the same time an optimal solution of the primal problem. We also consider a variant of this new algorithm, based on scaling the “dual” parametric function. The numerical results, in case of quadratic-linear ratios and linear constraints, show that the performance of the new algorithm and its scaled version is superior to that of the Dinkelbach-type algorithms. From the computational results it also appears that contrary to the primal approach, the “dual” approach is less influenced by scaling.
12742018
On Cones of Nonnegative Quadratic Functions
We derive linear matrix inequality (LMI) characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co-positivity. These matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain. As a domain, we consider for instance the intersection of a (upper) level-set of a quadratic function and a half-plane. Consequently, we arrive at a generalization of Yakubovich's S-procedure result. Although the primary concern of this paper is to characterize the matrix cones by LMIs, we show, as an application of our results, that optimizing a general quadratic function over the intersection of an ellipsoid and a half-plane can be formulated as semidefinite programming (SDP), thus proving the polynomiality of this class of optimization problems, which arise, e.g., from the application of the trust region method for nonlinear programming. Other applications are in control theory and robust optimization.
8339552
A sequence-based filtering method for ncRNA identification and its application to searching for riboswitch elements
MOTIVATION
Recent studies have uncovered an "RNA world", in which non coding RNA (ncRNA) sequences play a central role in the regulation of gene expression. Computational studies on ncRNA have been directed toward developing detection methods for ncRNAs. State-of-the-art methods for the problem, like covariance models, suffer from high computational cost, underscoring the need for efficient filtering approaches that can identify promising sequence segments and speedup the detection process.


RESULTS
In this paper we make several contributions toward this goal. First, we formalize the concept of a filter and provide figures of merit that allow comparison between filters. Second, we design efficient sequence based filters that dominate the current state-of-the-art HMM filters. Third, we provide a new formulation of the covariance model that allows speeding up RNA alignment. We demonstrate the power of our approach on both synthetic data and real bacterial genomes. We then apply our algorithm to the detection of novel riboswitch elements from the whole bacterial and archaeal genomes. Our results point to a number of novel riboswitch candidates, and include genomes that were not previously known to contain riboswitches.


AVAILABILITY
The program is available upon request from the authors.