12615212
Finite locally-quasiprimitive graphs
A finite graph is said to be locally-quasiprimitive relative to a subgroup G of automorphisms if, for all vertices α, the stabiliser in G of α is quasiprimitive on the set of vertices adjacent to α. (A permutation group is said to be quasiprimitive if all of its non-trivial normal subgroups are transitive.) The graph theoretic condition of local quasiprimitivity is strictly weaker than the conditions of local primitivity and 2-arc transitivity which have been studied previously. It is shown that the family of locally-quasiprimitive graphs is closed under the formation of a certain kind of quotient graph, called a normal quotient, induced by a normal subgroup. Moreover, a locally-quasiprimitive graph is proved to be a multicover of each of its normal quotients. Thus finite locally-quasiprimitive graphs which are minimal in the sense that they have no non-trivial proper normal quotients form an important sub-family, since each finite locally-quasiprimitive graph has at least one such graph as a normal quotient. These minimal graphs in the family are called "basic" locally-quasiprimitive graphs, and their structure is analysed. The process of constructing locally-quasiprimitive graphs with a given locally-quasiprimitive graph Σ as a normal quotient is then considered. It turns out that this can be viewed as a problem of constructing covering graphs of certain multigraphs associated with Σ. Further, it is shown that, under certain conditions, a locally-quasiprimitive graph can be reconstructed from knowledge of two of its normal quotients. Finally a series of open problems is presented.
38623555
Domination number and neighbourhood conditions
Abstract The domination number γ of a graph G is the minimum cardinality of a subset D of vertices of G such that each vertex outside D is adjacent to at least one vertex in D . For any subset A of the vertex set of G , let ∂ + ( A ) be the set of vertices not in A which are adjacent to at least one vertex in A . Let N (A) be the union of A and ∂ + ( A ), and d ( A ) be the sum of degrees of all the vertices of A . In this paper we prove the inequality 2q ⩽ (p−γ)(p−γ+2)−|∂ + (A)| (p−γ+1)+d( N (A)) , and characterize the extremal graphs for which the equality holds, where p and q are the numbers of vertices and edges of G , respectively. From this we then get an upper bound for γ which generalizes the known upper bound γ⩽p+1− 2q+1 . Let I ( A ) be the set of vertices adjacent to all vertices of A , and I (A) be the union of A and I ( A ). We prove that 2q ⩽ (p − γ − | I (A)| + 2)(p − γ + 4) + d( I (A)) − min {p − γ − | I (A)| + 2, |A|, |I(A)|, 3} , which implies an upper bound for γ as well.
20645165
Harper-type lower bounds and the bandwidths of the compositions of graphs
Abstract We give a general Harper-type lower bound for the bandwidth of a graph which is a common generalization of several known results. As applications we get a lower bound for the bandwidth of the composition of two graphs. By using this we determine the bandwidths of some composition graphs such as ( P r × P s )[ H ], ( P r × C s )[ H ] (2 r ≠ s ), ( C r × C s )[ H ] (6 ⩽ 2 r ⩽ s ), etc., for any graph H . Interestingly, the bandwidths of these graphs have nothing to do with the structure of H in general.
8520561
Bounding the bandwidths for graphs
Let G;H be nite graphs with jV(H)j>jV(G)j. The bandwidth of G with respect to H is dened to be BH(G) = minmaxuv2E(G)dH((u);(v)), with the minimum taken over all injections from V(G )t oV(H), where dH(x;y) is the distance in H between two vertices x;y 2 V(H). This number is involved with the VLSI design and optimization, especially when the \host" graph H is a path Pn or a cycle Cn of length n = jV(G)j. In these two cases, BH(G) is known to be the ordinary bandwidth B(G) and the cyclic bandwidth Bc(G), respectively, and the corresponding decision problem is NP-complete. So estimations of B(G), Bc(G) and in general BH(G) are needed, especially in determining the bandwidths of some specic graphs. In this paper, we rst propose a systematic method for obtaining lower bounds for the bandwidth BH(G). By using this method, we then get a number of lower bounds for B(G) and Bc(G )i n terms of some distance- and degree-related parameters. c 2000 Elsevier Science B.V. All rights reserved.
34996713
Polynomial Time Solvability Of The Weighted Ring Arc-Loading Problem With Integer Splitting
In the Weighted Ring Arc-Loading Problem with Integer Splitting, we are given a set of communication requests each associated with a non-negative integer weight. The problem is to find a routing scheme such that the maximum load on arcs of the ring is minimized, subject to that the weight of each request may be split into two integral parts routed in two directions around the ring, where the load of an arc is the sum of parts routed through the arc. A pseudo-polynomial algorithm for this problem is implied by a result in [G. Wilfong and P. Winkler, Ring routing and wavelength translation, Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms, San Fancisco, CA, 1998, 333-341]. By refining the rounding technique developed in the same paper, we prove that the problem can be solved in polynomial time.
14537383
Constructing a Class of Symmetric Graphs
We find a natural construction of a large class of symmetric graphs from point- and block-transitive 1-designs. The graphs in this class can be characterized as G -symmetric graphs whose vertex sets admit a G -invariant partition B of block size at least 3 such that, for any two blocks B, C of B, either there is no edge between B and C, or there exists only one vertex in B not adjacent to any vertex inC . The special case where the quotient graph ?Bof ? relative to B is a complete graph occurs if and only if the 1-design needed in the construction is a G -doubly transitive and G -block-transitive 2-design, and in this case we give an explicit classification of ? when G is a doubly transitive projective group or an affine group containing the affine general group. Examples of such graphs include cross ratio graphs studied recently by Gardiner, Praeger and Zhou and some other graphs with vertices the (point, line)-flags of the projective or affine geometry.
61202442
Probabilistic identity characterization for face recognition
We present a general framework for characterizing the object identity in a single image or a group of images with each image containing a transformed version of the object, with applications to face recognition. In terms of the transformation, the group is made of either many still images or frames of a video sequence. The object identity is either discrete- or continuous-valued. This probabilistic framework integrates all the evidence of the set and handles the localization problem, illumination and pose variations through subspace identity encoding. Issues and challenges arising in this framework are addressed and efficient computational schemes are presented. Good face recognition results using the PIE database are reported.
18318661
Characterization of Human Faces under Illumination Variations Using Rank, Integrability, and Symmetry Constraints
Photometric stereo algorithms use a Lambertian reflectance model with a varying albedo field and involve the appearances of only one object. This paper extends photometric stereo algorithms to handle all the appearances of all the objects in a class, in particular the class of human faces. Similarity among all facial appearances motivates a rank constraint on the albedos and surface normals in the class. This leads to a factorization of an observation matrix that consists of exemplar images of different objects under different illuminations, which is beyond what can be analyzed using bilinear analysis. Bilinear analysis requires exemplar images of different objects under same illuminations. To fully recover the class-specific albedos and surface normals, integrability and face symmetry constraints are employed. The proposed linear algorithm takes into account the effects of the varying albedo field by approximating the integrability terms using only the surface normals. As an application, face recognition under illumination variation is presented. The rank constraint enables an algorithm to separate the illumination source from the observed appearance and keep the illuminant-invariant information that is appropriate for recognition. Good recognition results have been obtained using the PIE dataset.
728
Analyzing and modelling the AS-level Internet topology
Recently we introduced the rich-club phenomenon as a quantitative metric to characterize the tier structure of the Autonomous Systems level Internet topology (AS graph) and we proposed the Interactive Growth (IG) model, which closely matches the degree distribution and hierarchical structure of the AS graph and compares favourble with other available Internet power-law topology generators. Our research was based on the widely used BGP AS graph obtained from the Oregon BGP routing tables. Researchers argue that Traceroute AS graph, extracted from the traceroute data collected by the CAIDA's active probing tool, Skitter, is more complete and reliable. To be prudent, in this paper we analyze and compare topological structures of Traceroute AS graph and BGP AS graph. Also we compare with two synthetic Internet topologies generated by the IG model and the well-known Barabasi-Albert (BA) model. Result shows that both AS graphs show the rich-club phenomenon and have similar tier structures, which are closely matched by the IG model, however the BA model does not show the rich-club phenomenon at all.
11541594
A network-centric approach to enhancing the interactivity of large-scale distributed virtual environments
In Distributed Virtual Environments (DVEs), where many simultaneous human users interact with each other in a shared, 3D virtual world, enhancing interactivity is of crucial importance, since the success of a DVE greatly depends on users' perceptions. In general, enhancing interactivity for large-scale DVEs with thousands of geographically distributed users faces various challenges such as large Internet latencies, high resource demands and high potential of security threats. In this paper, we propose a network-centric approach to enhance the interactivity of DVEs by directly minimizing the network latencies in client-server communications. We consider two key problems: the server placement problem and the client assignment problem. We have formulated these two problems and proved that they are both NP-hard. We then adapt and develop several degree-based server placement approaches and greedy client assignment algorithms. Extensive simulation studies using realistic models have shown that our approach is very effective in enhancing the interactivity of large-scale DVEs.
9649594
Probabilistic Human Recognition from Video
This paper presents a method for incorporating temporal information in a video sequence for the task of human recognition. A time series state space model, parameterized by a tracking state vector and a recognizing identity variable, is proposed to simultaneously characterize the kinematics and identity. Two sequential importance sampling (SIS) methods, a brute-force version and an efficient version, are developed to provide numerical solutions to the model. The joint distribution of both state vector and identity variable is estimated at each time instant and then propagated to the next time instant. Marginalization over the state vector yields a robust estimate of the posterior distribution of the identity variable. Due to the propagation of identity and kinematics, a degeneracy in posterior probability of the identity variable is achieved to give improved recognition. This evolving behavior is characterized using changes in entropy. The effectiveness of this approach is illustrated using experimental results on low resolution face data and upper body data.