Signed and minus clique-transversal functions on graphs

A minus (respectively, signed) clique-transversal function of a graph G = ( V , E ) is a function f : V → { − 1 , 0 , 1 } (respectively, { − 1 , 1 } ) such that ∑ u ∈ C f ( u ) ⩾ 1 for every maximal clique C of G. The weight of a minus (respectively, signed) clique-transversal function of G is f ( V ) = ∑ v ∈ V f ( v ) . The minus (respectively, signed) clique-transversal problem is to find a minus (respectively, signed) clique-transversal function of G of minimum weight. In this paper, we present a unified approach to these two problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. We also prove that the signed clique-transversal problem is NP-complete for chordal graphs and planar graphs.