Vertex-edge Roman domination in graphs: complexity and algorithms

For a simple, undirected graph G(V,E), a function h:V(G)→{0,1,2} such that each edge (u,v) of G is either incident with a vertex with weight at least one or there exists a vertex w such that either (u,w)∈E(G) or (v,w)∈E(G) and h(w)=2, is called a vertex-edge Roman dominating function (ve-RDF) of G. For a graph G, the smallest possible weight of a ve-RDF of G which is denoted by γveR(G), is known as the extit{vertex-edge Roman domination number} of G. The problem of determining γveR(G) of a graph G is called minimum vertex-edge Roman domination problem (MVERDP). In this article, we show that the problem of deciding if G has a ve-RDF of weight at most l for star convex bipartite graphs, comb convex bipartite graphs, chordal graphs and planar graphs is NP-complete. On the positive side, we show that MVERDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, a 2-approximation algorithm for MVERDP is presented. It is also shown that vertex cover and vertex-edge Roman domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MVERDP is presented.