Algorithmic complexity of weakly connected Roman domination in graphs

For a simple, undirected, connected graph G, a function h: V(G) -> {0, 1, 2} which satisfies the following conditions is called a weakly connected Roman dominating function (WCRDF) of G with weight h(V) = Sigma(p is an element of V) h(p). (C1). For all q is an element of V with h(q) = 0 there exists a vertex r such that (q, r) is an element of E and h(r) = 2 and (C2). The graph with vertex set V(G) and edge set {(p, z) : h(p) >= 1 or h(z) >= 1 or both} 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 2(1 + epsilon)(1 +In(Delta - 1))-approximation algorithm for the MWCRDP and show that the same cannot have (1 - delta) In vertical bar V vertical bar ratio approximation algorithm for any delta > 0 unless P = NP. Next, we show that MWCRDP is APX-hard for graphs with Delta = 4. 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.