Trapezoid graphs are a class of cocomparability graphs containing interval graphs and permutation graphs as subclasses. They were introduced by Dagan et al. [3]. They propose an 0(n2) algorithm for chromatic number and a less efficient algorithm for maximum clique on trapezoid graphs. Based on a geometric representation of trapezoid graphs by boxes in the plane we design optimal, i.e., O(n log ...

We study the complexity of a group of distance-constrained graph labeling problems when parameterized by the neighborhood diversity (nd), which is a natural graph parameter between vertex cover and clique width. Neighborhood diversity has been used to generalize and speed up FPT algorithms previously parameterized by vertex cover, as is also demonstrated by our paper. We show that the Uniform C...

Cardinality Maximum Flow Network Interdiction Problem (CMFNIP) is known to be strongly NP-hard problem in the literature. A particular case of CMFNIP has been shown to have reduction from clique problem. In the present work,an effort is being made to solve this particular case of CMFNIP in polynomial time. Direct implication of this solution is that the clique problem gets solved in polynomial ...

In a paper published in Journal of Combinatorial Theory, Series B (1986), Monma and Wei propose an extensive study of the intersection graph of paths on a tree. They explore this notion by varying the notion of intersection: the paths are respectively considered to be the sets of their vertices and the sets of their edges, and the trees may or may not be directed. Their main results are a chara...

Description Given an undirected graph G = ( V, E ), a clique S is a subset of V such that for any two elements u, v ∈ S, ( u, v ) ∈ E. Using the notation ES to represent the subset of edges which have both endpoints in clique S, the induced graph GS = ( S, ES ) is complete. Finding the largest clique in a graph is an NP-hard problem, called the maximum clique problem (MCP). Cliques are intimate...

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...

Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS’09) and of Fomin et al. (FOCS’10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use a structural parameters of the inp...

Let S be a finite, nonempty set of positive integers. Then, the divisor graph ) (S D of S has S as its vertex set, and vertices i and j are adjacent if and only if either j i or i j . This paper investigates the vertex-chromatic number, the clique number, the clique cover number, and the independence number of ]) ([n D and its complement, where } , 1 : { ] [ Ν ∈ ≤ ≤ = n n i i n . Besides, we di...