Algorithmic aspects of Roman domination in graphs

Abstract

Let G be a simple, undirected graph. A function g : V (G) → {0, 1, 2, 3} having the property that v∈N G (u) g(v) ≥ 3, if g(u) = 0, and v∈N G (u) g(v) ≥ 2, if g(u) = 1 for any vertex u ∈ G, where NG(u) is the set of vertices adjacent to u in G, is called a Roman {3}-dominating function (R3DF) of G. The weight of a R3DF g is the sum g(V ) = v∈V g(v). The minimum weight of a R3DF is called the Roman {3}-domination number and is denoted by γ {R3} (G). Given a graph G and a positive integer k, the Roman {3}-domination problem (R3DP) is to check whether G has a R3DF of weight at most k. In this paper, first we show that the R3DP is NP-complete for chordal graphs, planar graphs and for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. The minimum Roman {3}-domination problem (MR3DP) is to find a R3DF of minimum weight in the input graph. We show that MR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. We propose a 3(1 + ln(∆ − 1))-approximation algorithm for the MR3DP, where ∆ is the maximum degree of G and show that the MR3DP problem cannot be approximated within (1 − ϵ) ln |V | for any ϵ > 0 unless N P ⊆ DT IM E(|V | O(log log |V |) ). Next, we show that the MR3DP problem is APX-complete for graphs with maximum degree 4. We also show that the domination and Roman {3}-domination problems are not equivalent in computational complexity aspects. Finally, an ILP formulation for MR3DP is proposed.