Algorithmic Aspects of Total Roman and Total Double Roman Domination in Graphs

Abstract

For a simple, undirected and connected graph \(G = (V, E)\), a total Roman dominating function (TRDF) \(f : V \rightarrow \lbrace 0, 1, 2 \rbrace \) has the property that, every vertex u with \(f(u) = 0\) is adjacent to at least one vertex v for which \(f(v) = 2\) and the subgraph induced by the set of vertices labeled one or two has no isolated vertices. A total double Roman dominating function (TDRDF) on G is a function \(f : V \rightarrow \lbrace 0, 1, 2, 3 \rbrace \) such that for every vertex \(v \in V\) if \(f(v) = 0\), then v has at least two neighbors x, y with \(f(x) = f(y) = 2\) or one neighbor w with \(f(w) = 3\), and if \(f(v) = 1\), then v must have at least one neighbor w with \(f(w) \ge 2\) and the subgraph induced by the set \(\{u_i : f(u_i) \ge 1\}\) has no isolated vertices. The weight of a T(D)RDF f is the sum \(f(V) = \sum _{v \in V}f(v)\). The minimum total (double) Roman domination problem (MT(D)RDP) is to find a T(D)RDF of minimum weight of the input graph. In this article, we show that MTRDP and MTDRDP are polynomial time solvable for bounded treewidth graphs, chain graphs and threshold graphs. We design a \(2 (\ln (\varDelta - 0.5) + 1.5)\)-approximation algorithm (APX-AL) for the MTRDP and \(3 (\ln (\varDelta - 0.5) + 1.5)\)-APX-AL for the MTDRDP, where \(\varDelta \) is the maximum degree of G, and show that the same cannot have \((1 - \delta ) \ln |V|\) ratio APX-AL for any \(\delta > 0\) unless \(P = NP\). Finally, we show that MT(D)RDP is APX-hard for graphs with \( \varDelta =5\).KeywordsTotal Roman dominationTotal double Roman dominationAPX-complete