We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this paper we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable. Moreover, given a B1-EPG repres...

We define an atom tree of a graph as a generalization of a clique tree: its nodes are the atoms obtained by clique minimal separator decomposition, and its edges correspond to the clique minimal separators of the

Clique topology is a central theme in this paper and is best explained by breaking it down into two terms. A clique is a complete graph, C that is a subgraph of a directed graph G = (V,E). Alternatively, a clique in an undirected graph G is a subset of vertices, C ⊆ V where every pair of distinct vertices are adjacent. Topology in this case refers to the study of the orientation of clique conne...

In this paper we give the first characterisation of graphs with linear clique-width at most 3, and we give a polynomial-time recognition algorithm for such graphs. In addition, we give a new characterisation of graphs with linear clique-width at most 2 and a new layout characterisation of linear clique-width in general. Among our results is also a decomposition scheme that preserves the linear ...

Suppose G=( V,E) is a graph in which each maximal clique Ci is associated with an integer ri, where 0 < ri < ICi 1. The generalized clique transversal problem is to determine the minimum cardinality of a subset D of V such that ID n Cij >ri for every maximal clique Ci of G. The problem includes the clique-transversal problem, the i, 1 clique-cover problem, and for perfect graphs, the maximum q-...

An i-triangulated graph is a graph in which every odd cycle has two non-crossing chords; i-triangulated graphs form a subfamily of perfect graphs. A slightly more general family of perfect graphs are clique-separable graphs. A graph is clique-separable precisely if every induced subgraph either has a clique cutset, or is a complete multipartite graph or a clique joined to an arbitrary bipartite...

We study the linear clique-width of graphs that are obtained from paths by disjoint union and adding true twins. We show that these graphs have linear clique-width at most 4, and we give a complete characterisation of their linear clique-width by forbidden induced subgraphs. As a consequence, we obtain a linear-time algorithm for computing the linear clique-width of the considered graphs. Our r...

We show that the edge-clique graphs of cocktail party graphs have unbounded rankwidth. This, and other observations lead us to conjecture that the edge-clique cover problem is NP-complete for cographs. We show that the independent set problem on edge-clique graphs of cographs and of distance-hereditary graphs can be solved in O(n4) time. We show that the independent set problem on edge-clique g...