Algorithmic complexity of weakly connected Roman domination in graphs

Abstract

For a simple, undirected, connected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called a weakly connected Roman dominating function (WCRDF) of [Formula: see text] with weight [Formula: see text]. (C1). For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text] and (C2). The graph with vertex set [Formula: see text] and edge set [Formula: see text] or [Formula: see text] or both[Formula: see text] is connected. The problem of determining WCRDF of minimum weight is called minimum weakly connected Roman domination problem (MWCRDP). In this paper, we show that MWCRDP is polynomial time solvable for bounded treewidth graphs, threshold graphs and chain graphs. We design a [Formula: see text]-approximation algorithm for the MWCRDP and show that the same cannot have [Formula: see text] ratio approximation algorithm for any [Formula: see text] unless [Formula: see text]. Next, we show that MWCRDP is APX-hard for graphs with [Formula: see text]. We also show that the domination and weakly connected Roman domination problems are not equivalent in computational complexity aspects. Finally, two different integer linear programming formulations for MWCRDP are proposed.