Vertex-Edge Roman Domination in Graphs: Complexity and Algorithms

Abstract

For a simple, undirected graph $G(V,E)$, a function $h : V(G) rightarrow lbrace 0, 1, 2rbrace$ 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 $(u,w) in E(G)$ or $(v,w) in E(G)$ 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 $gamma_{veR}(G)$, is known as the textit{vertex-edge Roman domination number} of $G$. The problem of determining $gamma_{veR}(G)$ 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.