Algorithmic aspects of total Roman ${2}$-domination in graphs

Abstract

For a simple, undirected, connected graph G, a function h : V → {0, 1, 2} is called a total Roman {2}-dominating function (TR2DF) if for every vertex v in V with weight 0, either there exists a vertex u in NG(v) with weight 2, or at least two vertices x, y in NG(v) each with weight 1, and the subgraph induced by the vertices with weight more than zero has no isolated vertices. The weight of TR2DF h is ∑peV h(p). The problem of determining TR2DF of minimum weight is called minimum total Roman {2}-domination problem (MTR2DP). We show that MTR2DP is polynomial time solvable for bounded treewidth graphs, threshold graphs and chain graphs. We design a 2(ln(Δ - 0.5) + 1.5)-approximation algorithm for the MTR2DP and show that the same cannot have (1-δ) ln(|V|) ratio approximation algorithm for any δ > 0 unless P=Np . Next, we show that MTR2DP is APX-hard for graphs with Δ=4. We also show that the domination and TR2DF problems are not equivalent in computational complexity aspects.