A clique covering of a graph G is a set of cliques of G in which each edge of G is contained in at least one clique. The smallest cardinality of clique coverings of G is called the clique covering number of G. A glued graph results from combining two nontrivial vertex-disjoint graphs by identifying nontrivial connected isomorphic subgraphs of both graphs. Such subgraphs are referred to as the c...

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. A graph G is clique-perfect if the sizes of a minimum clique-transversal and a maximum clique-independent set are equal for every induced subgraph ofG. The list of minimal forbidden induced subgraphs for the class of clique-perf...

The clique partitioning problem (CPP) can be formulated as follows. Given is a complete graph G = (V; E), with edge weights w ij 2 R for all fi; jg 2 E. A subset A E is called a clique partition if there is a partition of V into non-empty, disjoint sets V 1 S k p=1 ffi; jgji; j 2 V p g. The weight of such a clique partition A is deened as P fi;jg2A w ij. The problem is now to nd a clique partit...

MAXIMUM CLIQUE PROBLEM the most relevant problem in Graph theory, known for years still do not have its polynomial time solution. Many algorithms have been proposed, still the problem lie the same i.e. to find the Clique in the polynomial time. The Clique problem is to figure out the sub graph with the maximum cardinality. Maximum clique in a random graph is NP-Hard problem, actuated by many pr...

A graph has linear clique-width at most k if it has a clique-width expression using at most k labels such that every disjoint union operation has an operand which is a single vertex graph. We give the first characterisation of graphs of linear clique-width at most 3, and we give the first polynomial-time recognition algorithm for graphs of linear clique-width at most 3. In addition, we present ...

The clique graph of a graph G is the intersection graph K(G) of the (maximal) cliques of G. A graph G is called self-clique whenever G ∼= K(G). This paper gives various constructions of self-clique graphs. In particular, we employ (r, g)-cages to construct self-clique graphs whose set of clique-sizes is any given finite set of integers greater than 1.

The clique graph of G, K(G), is the intersection graph of the family of cliques (maximal complete sets) of G. Clique-critical graphs were defined as those whose clique graph changes whenever a vertex is removed. We prove that if G has m edges then any clique-critical graph in K−1(G) has at most 2m vertices, which solves a question posed by Escalante and Toft [On clique-critical graphs, J. Combi...

Given a graph G and an interdiction budget k∈N, the Edge Interdiction Clique Problem (EICP) asks to find subset of at most k edges remove from so that size maximum clique, in interdicted graph, is minimized. The EICP belongs family problems with aim reducing clique number graph. optimal solutions, called policies, determine vital which are crucial for preserving its number. We propose new set-c...

1. ABSTRACT Cliques refer to subgraphs in an undirected graph such that vertices in each subgraph are pairwise adjacent. The maximum clique problem, to find the clique with most vertices in a given graph, has been extensively studied. Besides its theoretical value as an NPhard problem, the maximum clique problem is known to have direct applications in various fields, such as community search in...

Multi-clique-width is obtained by a simple modification in the definition of cliquewidth. It has the advantage of providing a natural extension of tree-width. Unlike clique-width, it does not explode exponentially compared to tree-width. Efficient algorithms based on multi-clique-width are still possible for interesting tasks like computing the independent set polynomial or testing c-colorabili...