This column addresses the following questions about the complexity of approximating the CLIQUE problem. 1. Is nding the vertices of the largest clique more diicult than merely nding the vertices of a 2-approximate clique (that is, a clique with at least half the size of the largest clique)? 2. What computational resources can be used to nd the vertices of a 2-approximate clique? 3. Could fewer ...

This paper studies the maximum-clique independence problem and some variations of clique transversal such as {k}-clique, maximum-clique, minus clique, signed k-fold problems from algorithmic aspects for k-trees, suns, planar graphs, doubly chordal perfect total split line dually graphs. We give equations to compute numbers show that {k}-clique is polynomial-time solvable graphs whose equal thei...

We discuss the complexity of computing various graph polynomials of graphs of fixed clique-width. We show that the chromatic polynomial, the matching polynomial and the two-variable interlace polynomial of a graph G of clique-width at most k with n vertices can be computed in time O(n), where f(k) ≤ 3 for the inerlace polynomial, f(k) ≤ 2k + 1 for the matching polynomial and f(k) ≤ 3 · 2 for th...

In this paper we compare and illustrate the algorithmic use of graphs of bounded treewidth and graphs of bounded clique-width. For this purpose we give polynomial time algorithms for computing the four basic graph parameters independence number, clique number, chromatic number, and clique covering number on a given tree structure of graphs of bounded tree-width and graphs of bounded clique-widt...

A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs some problems that are NP-complete on general graphs, such as the maximum...

We show that #SAT is polynomial-time tractable for classes of CNF formulas whose incidence graphs have bounded symmetric clique-width (or bounded clique-width, or bounded rank-width). This result strictly generalizes polynomial-time tractability results for classes of formulas with signed incidence graphs of bounded clique-width and classes of formulas with incidence graphs of bounded modular t...

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...

The chromatic polynomial gives the number of proper λ-colourings of a graph G. This paper considers factorisation of the chromatic polynomial as a first step in an algebraic study of the roots of this polynomial. The chromatic polynomial of a graph is said to have a chromatic factorisation if P (G, λ) = P (H1, λ)P (H2, λ)/P (Kr, λ) for some graphs H1 and H2 and clique Kr. It is known that the c...

The maximum clique problem is known to be a typical NP-complete problem, and hence it is believed to be impossible to solve it in polynomial-time. So, it is important to know a reasonable sufficient condition under which the maximum clique problem can be proved to be polynomial-time solvable. In this paper, given a graph of vertices and whose maximum degree is , we prove that if is less than or...

A characteristic polynomial was recently de!ned for greedoids, generalizing the notion for matroids. When chordal graphs are viewed as antimatroids by shelling of simplicial vertices, the greedoid characteristic polynomial gives additional information about those graphs. In particular, the characteristic polynomial for a chordal graph is an alternating clique generating function and is expressi...