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.