Given a graph G = (V, E) and a positive integer k, the PARTITION INTO CLIQUES (PIC) decision problem consists of deciding whether there exists a partition of V into k disjoint subsets V1, V2, . . . , Vk such that the subgraph induced by each part Vi is a complete subgraph (clique) of G. In this paper, we establish both the NP-completeness of PIC for planar cubic graphs and the Max SNP-hardness ...

We define a multivariate polynomial that generalizes in a unified way the twovariable interlace polynomial defined by Arratia, Bollobás and Sorkin on the one hand, and a one-variable variant of it defined by Aigner and van der Holst on the other. We determine a recursive definition for our polynomial that is based on local complementation and pivoting like the recursive definitions of Tutte’s p...

The proofs of major results of Computability Theory like Rice, Rice-Shapiro or Kleene’s fixed point theorem hide more information of what is usually expressed in their respective statements. We make this information explicit, allowing to state stronger, complexity theoretic-versions of all these theorems. In particular, we replace the notion of extensional set of indices of programs, by a set o...

A circular-arc graph is the intersection graph of arcs on a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. A clique-independent set of a graph is a set of pairwise disjoint cliques of the graph. It is NP-hard to compute the maximum cardinality of a clique-independent set for a general graph. In the present paper, we propose po...

Given a graph G whose adjacency matrix is A, the Motzkin-Strauss formulation of the Maximum-Clique Problem is the quadratic program maxfx T Axjx T e = 1; x 0g. It is well known that the global optimum value of this QP is (1 ? 1=!(G)), where !(G) is the clique number of G. Here, we characterize the following: 1) rst order optimality 2) second order optimality 3) local optimality 4) strict local ...

A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. It is NP-hard to determine the minimum cardinality τc of a clique-transversal of G. In this work, first we propose an algorithm for determining this parameter for a general graph, which runs in polynomial time, for fixed τc. This algorithm is employed for finding the minimum cardinality clique-transvers...

Given a graph G whose adjacency matrix is A, the Motzkin-Strauss formulation of the Maximum-Clique Problem is the quadratic program maxfxTAxjxT e = 1; x 0g. It is well known that the global optimum value of this QP is (1 1=!(G)), where !(G) is the clique number of G. Here, we characterize the following: 1) rst order optimality 2) second order optimality 3) local optimality 4) strict local. Thes...

A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size b19n/30 + 2/15c. This bound is tight, since 19n/30− 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connecte...

In a directed graph, a kernel is a subset of vertices that is both stable and absorbing. Not all digraphs have a kernel, but a theorem due to Boros and Gurvich guarantees the existence of a kernel in every clique-acyclic orientation of a perfect graph. However, an open question is the complexity status of the computation of a kernel in such a digraph. Our main contribution is to prove new polyn...

Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of following problems by k: (1) Given a graph G, clique modulator D (a is set vertices, whose removal results in clique) size k for list L(v) colors every v ∈ V(G), decide whether G has proper coloring; (2) pre-coloring λ_P: X → Q ⊆ λ_P can be extended coloring using only from Q. For Problem 1 we des...