The algorithmic complexity of the minus clique-transversal problem

A minus clique-transversal function of a graph G = (V, E) is a function f : V → {- 1, 0, 1} such that ∑u ∈ V (C) f (u) ≥ 1 for every clique C of G. The weight of a minus clique-transversal function is f (V) = ∑ f (v), over all vertices v ∈ V. The minus clique-transversal number of a graph G, denoted τc - (G), equals the minimum weight of a minus clique-transversal function of G. The upper minus clique-transversal number of a graph G, denoted Tc - (G), equals the maximum weight of a minimal minus clique-transversal function of G. In this paper, we show that the minus clique-transversal problem (respectively, upper minus clique-transversal problem) is NP-complete even when restricted to chordal graphs. On the other hand, we give a linear algorithm to solve the minus clique-transversal problem for block graphs.Department of Logistics and Maritime Studie