In this paper, we find lower bounds for the maximum and minimum numbers of cliques in maximal sets of pairwise disjoint cliques in a graph . By complementation, these yield lower bounds for the maximum and minimum numbers of independent sets in maximal sets of pairwise disjoint maximal independent sets of vertices in a graph . In the latter context, we show by examples that one of our bounds is...

Let L be a finite set associated with cliques of a distance-regular graph of order (s, t), with d-cliques of Johnson graphs and antipodal distance-regular graphs of diameter d, respectively. If we partially order L by the ordinary inclusion, three families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials. c © 2007 E...

In this paper we address the problem of finding all maximal cliques in a subclass of circular-arc graphs. We consider only circular-arc graphs that are generated with an arc model where no three arcs cover the whole circle. We prove that the number of maximal cliques is bounded by the number of vertices and that for each maximal clique it exists a point in the circle which is covered by all arc...

In this paper, we prove that for any forest F ⊂ Kn, the edges of E(Kn)\E(F ) can be partitioned into O(n log n) cliques. This extends earlier results on clique partitions of the complement of a perfect matching and of a hamiltonian path in Kn. We also show that if a graph G has maximum degree 4, then the edges of E(Kn)\E(G) can be partitioned into roughly n 3 24 1 2 log n cliques provided there...

Exact exponential-time algorithms for NP-hard problems is an emerging field, and an increasing number of new results are being added continuously. Two important NP-hard problems that have been studied for decades are the treewidth and the minimum fill problems. Recently, an exact algorithm was presented by Fomin, Kratsch, and Todinca to solve both of these problems in time O∗(1.9601n). Their al...

A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The list of minimal forbidden induced subgraphs for the c...

In many applications of computer vision and pattern recognition which use graph-based knowledge representation, it is of great interest to be able to extract the K largest cliques in a graph, but most methods are geared either towards extracting the single clique of maximum size, or enumerating all cliques, without following any particular order. In this paper we present a novel approach for pa...

In this paper, we introduce a non-associative higher-order graphical model to tackle the problem of semantic labeling of 3D point clouds. For this task, existing higher-order models overlook the relationships between the different classes and simply encourage the nodes in the cliques to have consistent labelings. We address this issue by devising a set of non-associative context patterns that d...

We will remove vertices from a given graph G of treewidth k in a good elimination ordering, and we can assume w.l.o.g. that G is connected. Some cliques in a current graph will be active cliques. For each active clique C and for each subset S of vertices in C, we store information if there exists a vertex cover of the original graph G that chooses exactly vertices in S out of the vertices in C....