The clique partitioning problem (CPP) can be formulated as follows: Given is a complete graph G = (V, E), with edge weights wij ∈ R for all {i, j} ∈ E. A subset A ⊆ E is called a clique partition if there is a partition of V into nonempty, disjoint sets V1, . . . , Vk , such that each Vp (p = 1, . . . ,k) induces a clique (i.e., a complete subgraph), and A = ∪ p=1 {{i, j}|i, j ∈ Vp , i ≠ j}. Th...

EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection graphs of paths on an orthogonal grid. The class Bk-EPG is the subclass of EPG graphs where the path on the grid associated to each vertex has at most k bends. Epstein et al. showed in 2013 that computing a maximum clique in B1-EPG graphs is polynomial. As remarked in [Heldt et al., 2014], when the number of bends is at le...

The problem of finding dense structures in a given graph is quite basic in informatics including data mining and data engineering. Clique is a popular model to represent dense structures, and widely used because of its simplicity and ease in handling. Pseudo cliques are natural extension of cliques which are subgraphs obtained by removing small number of edges from cliques. We here define a pse...

List partitions generalize list colourings and list homomorphisms. Each symmetric matrix M over 0; 1; deenes a list partition problem. Diierent choices of the matrix M lead to many well-known graph the-oretic problems including the problem of recognizing split graphs and their generalizations, nding homogeneous sets, joins, clique cutsets, stable cutsets, skew cutsets and so on. We develop tool...

In this paper we consider the class of simple graphs defined by excluding, as induced subgraphs, even holes (i.e., chordless cycles of even length) and diamonds (i.e., a graph obtained from a clique of size 4 by removing an edge). We say that such graphs are (evenhole, diamond)-free. For this class of graphs we first obtain a decomposition theorem, using clique cutsets, bisimplicial cutsets (wh...

Given two graphs G and H, assume that C = {C1, C2, . . . , Cq} is a clique cover of G and U is a subset of V (H). We introduce a new graph operation called the clique cover product, denoted by G ⋆ HU , as follows: for each clique Ci ∈ C , add a copy of the graph H and join every vertex of Ci to every vertex of U . We prove that the independence polynomial of G ⋆ HU I(G ⋆ H ;x) = I(H;x)I(G; xI(H...

Abstract. Finding the largest clique in random graphs is a well known hard problem. It is known that a random graph G(n, 1/2) almost surely has a clique of size about 2 log n. A simple greedy algorithm finds a clique of size log n, and it is a long-standing open problem to find a clique of size (1 + ǫ) log n in randomized polynomial time. In this paper, we study the generalization of finding th...

We construct a polynomial-time algorithm to approximate the branch-width of certain symmetric submodular functions, and give two applications. The first is to graph “clique-width”. Clique-width is a measure of the difficulty of decomposing a graph in a kind of tree-structure, and if a graph has clique-width at most k then the corresponding decomposition of the graph is called a “k-expression”. ...

The congested clique model of distributed computing has been receiving attention as a model for densely connected distributed systems. While there has been significant progress on the side of upper bounds, we have very little in terms of lower bounds for the congested clique; indeed, it is now known that proving explicit congested clique lower bounds is as difficult as proving circuit lower bou...

Two characterizations of intersection graphs of vertex disjoint paths in a tree, one in terms of maximal clique separator and the other in terms of minimal forbidden subgraphs, are presented. A polynomial recognition algorithm for this class is suggested.