Maximal Complete Subgraphs Research Articles

Clique Transversal Sets

Let G = (V, E) be a simple graph. A clique of G is a maximal complete subgraph of G. A subset D of V which meets every clique of G is called a clique transversal set of G. Since V(G) is a clique transversal set of G, the existence of a clique transversal set of G is guaranteed in any graph. The minimum cardinality of a clique transversal set of G is called the clique transversal number of G and is denoted by A study on this concept is initiated in this paper.

Dynamically Weighted Clique Evolution Model in Clique Networks

This paper proposes a weighted clique evolution model based on clique (maximal complete subgraph) growth and edge-weight driven for complex networks. The model simulates the scheme of real-world networks that the evolution of networks is likely to be driven by the flow, such as traffic or information flow needs, as well as considers that real-world networks commonly consist of communities. At each time step of a network’s evolution progress, an edge is randomly selected according to a preferential scheme. Then a new clique which contains the edge is added into the network while the weight of the edge is adjusted to simulate the flow change brought by the new clique addition. We give the theoretical analysis based on the mean field theory, as well as some numerical simulation for this model. The result shows that the model can generate networks with scale-free distributions, such as edge weight distribution and node strength distribution, which can be found in many real-world networks. It indicates that the evolution rule of the model may attribute to the formation of real-world networks.

Clique Matrix of a Graph in Traffic Control Problems

Clique matrix can be used as a graph theoretic tool to study traffic control problem at an intersection. A traffic control problem can be efficiently modelled as a graph where the nodes represent the traffic streams and the edges represent the relationship among the streams. The matrix representation of the control problem is used for phasing of signal groups and thus providing a solution to the control problem. Clique of a graph is defined as maximal complete subgraph and clique matrix of a graph is defined as a generalization of the incidence matrix where the columns correspond to the number of cliques and the rows correspond to the number of vertices of the graph.

Modeling Evolution of Weighted Clique Networks

We propose a weighted clique network evolution model, which expands continuously by the addition of a new clique (maximal complete sub-graph) at each time step. And the cliques in the network overlap with each other. The structural expansion of the weighted clique network is combined with the edges' weight and vertices' strengths dynamical evolution. The model is based on a weight-driven dynamics and a weights' enhancement mechanism combining with the network growth. We study the network properties, which include the distribution of vertices' strength and the distribution of edges' weight, and find that both the distributions follow the scale-free distribution. At the same time, we also find that the relationship between strength and degree of a vertex are linear correlation during the growth of the network. On the basis of mean-field theory, we study the weighted network model and prove that both vertices' strength and edges' weight of this model follow the scale-free distribution. And we exploit an algorithm to forecast the network dynamics, which can be used to reckon the distributions and the corresponding scaling exponents. Furthermore, we observe that mean-field based theoretic results are consistent with the statistical data of the model, which denotes the theoretical result in this paper is effective.

Bus transport network model with ideal [formula omitted]-depth clique network topology

We propose an ideal n-depth clique network model. In this model, the original network is composed of cliques (maximal complete subgraphs) that overlap with each other. The network expands continuously by the addition of new cliques. The final diameter of the network can be set in advance, namely, it is controllable. Assuming that the diameter of the network is n, the network exhibits a logistic structure with (n+1) layers. In this structure, the 0th layer represents the original network and each node of the (m)th layer (1≤m≤n) corresponds to a clique in the (m−1)th layer. In the growth process of the network, we ensure that any (m)th layer network is composed of overlapping cliques. Any node in an (m)th layer network corresponds to an m-depth community in the original network, and the diameter of an m-depth community is m. Therefore, the (n−1)th layer network will contain only one clique, the (n)th layer network will contain only one node, and the diameter of the corresponding original network is n. Then an ideal n-depth clique network will be obtained. Based on the ideal n-depth clique network model, we construct a bus transport network model with an ideal n-depth clique network topology (ICNBTN). Moreover, our study compares this model with the real bus transport network (RealBTN) of three major cities in China and a recently introduced bus transport network model (BTN) whose network properties correspond well with those of real BTNs. The network properties of the ICNBTN are much closer to those of the RealBTN than those of the BTN are. At the same time, the ICNBTN has higher clustering extent of bus routes, smaller network diameter, which corresponds to shorter maximum transfer times in a bus network, and lower average shortest path time coefficient than the BTN and the RealBTN. Therefore, the ICNBTN can achieve higher transfer efficiency for a bus transport system.

Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

Graphs and Algorithms A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.

Research on Some Bus Transport Networks with Random Overlapping Clique Structure

On the basis of investigating the statistical data of bus transport networks of three big cities in China, we propose that each bus route is a clique (maximal complete subgraph) and a bus transport network (BTN) consists of a lot of cliques, which intensively connect and overlap with each other. We study the network properties, which include the degree distribution, multiple edges' overlapping time distribution, distribution of the overlap size between any two overlapping cliques, distribution of the number of cliques that a node belongs to. Naturally, the cliques also constitute a network, with the overlapping nodes being their multiple links. We also research its network properties such as degree distribution, clustering, average path length, and so on. We propose that a BTN has the properties of random clique increment and random overlapping clique, at the same time, a BTN is a small-world network with highly clique-clustered and highly clique-overlapped. Finally, we introduce a BTN evolution model, whose simulation results agree well with the statistical laws that emerge in real BTNs.

Equivariant collapses and the homotopy type of iterated clique graphs

The clique graph K ( G ) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K 0 ( G ) = G , K n + 1 ( G ) = K ( K n ( G ) ) . We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results, it can be easily inferred that K n ( G ) is homotopy equivalent to G for every n if G belongs to the class of clique-Helly graphs or to the class of dismantlable graphs. However, in both of these cases the collection of iterated clique graphs is finite up to isomorphism. In this paper, we show two infinite classes of clique-divergent graphs that satisfy G ≃ K n ( G ) for all n, moreover K n ( G ) and G are simple-homotopy equivalent. We provide some results on simple-homotopy type that are of independent interest.

Dominant Sets and Pairwise Clustering

We develop a new graph-theoretic approach for pairwise data clustering which is motivated by the analogies between the intuitive concept of a cluster and that of a dominant set of vertices, a notion introduced here which generalizes that of a maximal complete subgraph to edge-weighted graphs. We establish a correspondence between dominant sets and the extrema of a quadratic form over the standard simplex, thereby allowing the use of straightforward and easily implementable continuous optimization techniques from evolutionary game theory. Numerical examples on various point-set and image segmentation problems confirm the potential of the proposed approach.

Graph relations, clique divergence and surface triangulations

AbstractThis work has two aims: first, we introduce a powerful technique for proving clique divergence when the graph satisfies a certain symmetry condition. Second, we prove that each closed surface admits a clique divergent triangulation. By definition, a graph is clique divergent if the orders of its iterated clique graphs tend to infinity, and the clique graph of a graph is the intersection graph of its maximal complete subgraphs. © 2005 Wiley Periodicals, Inc. J Graph Theory

Image Segmentation by Dominant Sets

Abstract We develop a framework for the image segmentation problem based on a new graph-theoretic formulation of clustering. The approach is motivated by the analogies between the intuitive concept of a cluster and that of a dominant set of vertices, a novel notion that generalizes that of a maximal complete subgraph to edge-weighted graphs. We also establish a correspondence between dominant sets and the extrema of a quadratic form over the standard simplex, thereby allowing us the use of continuous optimization techniques such as replicator dynamics from evolutionary game theory. Such systems are attractive as can easily be implemented in a parallel network of locally interacting computational units, and offer the advantage of biological plausibility. We present experimental results on real-world images which show the effectiveness of the approach.

The icosahedron is clique divergent

A clique of a graph G is a maximal complete subgraph. The clique graph k(G) is the intersection graph of the set of all cliques of G. The iterated clique graphs are defined recursively by k0(G)=G and kn+1(G)=k(kn(G)). A graph G is said to be clique divergent (or k-divergent) if limn→∞|V(kn(G))|=∞. The problem of deciding whether the icosahedron is clique divergent or not was (implicitly) stated Neumann-Lara in 1981 and then cited by Neumann-Lara in 1991 and Larrión and Neumann-Lara in 2000. This paper proves the clique divergence of the icosahedron among other results of general interest in clique divergence theory.

Maximal prime subgraph decomposition of Bayesian networks

The authors present a method for decomposition of Bayesian networks into their maximal prime subgraphs. The correctness of the method is proven and results relating the maximal prime subgraph decomposition (MPD) to the maximal complete subgraphs of the moral graph of the original Bayesian network are presented. The maximal prime subgraphs of a Bayesian network can be organized as a tree which can be used as the computational structure for LAZY propagation. We also identify a number of tasks performed on Bayesian networks that can benefit from MPD. These tasks are: divide and conquer triangulation, hybrid propagation algorithms combining exact and approximative inference techniques, and incremental construction of junction trees. We compare the proposed algorithm with standard algorithms for decomposition of undirected graphs into their maximal prime subgraphs. The discussion shows that the proposed algorithm is simpler, more easy to comprehend, and it has the same complexity as the standard algorithms.

A characterization of a topology on the countably infinite random graph

Let G be a countably infinite random graph and let M(G) be the collection of all maximal complete subgraphs of G with the subspace topology inherited from the power set 2G. We show that with probability 1, M(G) is homeomorphic to the irrationals.

Chordally signed graphs

Define a chordally signed graph to be a signed chordal graph (meaning that each edge is designated as being positive or negative and every induced cycle is a triangle) in which every positive cycle C (meaning every cycle C that contains an even number of negative edges) has a chord e such that C∪{ e} forms two positive cycles. Two characterizations of chordally signed graphs support the naturalness of this definition. In addition, chordally signed graphs can be easily recognized when the underlying graph has at most two maximal complete subgraphs (with minimal balancing vertex sets playing a key role).

Strong clique trees, neighborhood trees, and strongly chordal graphs

Maximal complete subgraphs and clique trees are basic to both the theory and applications of chordal graphs. A simple notion of strong clique tree extends this structure to strongly chordal graphs. Replacing maximal complete subgraphs with open or closed vertex neighborhoods discloses new relationships between chordal and strongly chordal graphs and the previously studied families of chordal bipartite graphs, clique graphs of chordal graphs (dually chordal graphs), and incidence graphs of biacyclic hypergraphs. © 2000 John Wiley & Sons, Inc. J. Graph Theory 33: 151–160, 2000

Locally C6 graphs are clique divergent

The clique graph kG of a graph G is the intersection graph of the family of all maximal complete subgraphs of G. The iterated clique graphs k n G are defined by k 0 G= G and k n+1 G= kk n G. A graph G is said to be k- divergent if V(k nG) tends to infinity with n. A graph is locally C 6 if the neighbours of any given vertex induce an hexagon. We prove that all locally C 6 graphs are k-divergent and that the diameters of the iterated clique graphs also tend to infinity with n while the sizes of the cliques remain bounded.

Some classes of strongly perfect graphs

A graph is perfect if for each of its induced subgraphs H, the chromatic number of H is equal to the maximum number of mutually adjacent vertices in it. A graph is ‘strongly perfect’ if each of its induced subgraphs H contains an independent set which meets all the cliques (maximal complete subgraphs) in it. Every strongly perfect graph is perfect but the converse is generally not valid. For example, the complement of an even cycle of length at least 6 is not strongly perfect though it is perfect. The strongly perfect graphs form an interesting class of perfect graphs, because the complement of a strongly perfect graph is not necessarily strongly perfect, unlike the case with the perfect graphs and their relevance to the famous Berge's strong perfect graph conjecture which has been eluding a solution for more than three decades. Meyniel graphs, line graphs that are free from some graphs, comparability graphs, costrongly perfect graphs are some of the most important classes of strongly perfect graphs. Here, we summarize the results concerning strongly perfect graphs.

A new characterization of strongly chordal graphs

The ‘strength’ of an edge or cycle is the number of maximal complete subgraphs it is in. Strongly chordal graphs are characterized by every cycle of edges of strengths at least k with no chord of strength at least k always itself having strength at least k. Among hereditary clique-Helly graphs, this is equivalent to the subgraph of edges of strength at least k always forming a chordal graph.

Clique divergent graphs with unbounded sequence of diameters

Abstract The clique graph kG of a graph G is the intersection graph of the family of all maximal complete subgraphs of G . The iterated clique graphs k n G are defined by k 0 G = G and k n +1 G = kk n G . A graph G is said to be k -divergent if | V ( k n G )| tends to infinity with n . We provide examples of k -divergent graphs such that the diameters of the iterated clique graphs also tend to infinity with n . Furthermore, the sizes of the cliques and even the chromatic numbers remain bounded.