Independent roman ${3}$-domination

Abstract

Let $G$ be a simple, undirected graph. In this paper, we initiate the study of independent Roman ${3}$-domination. A function $g : V(G) rightarrow lbrace 0, 1, 2, 3 rbrace$ having the property that $sum_{v in N_G(u)}^{} g(v) geq 3$, if $g(u) = 0$, and $sum_{v in N_G(u)}^{} g(v) geq 2$, if $g(u) = 1$ for any vertex $u in V(G)$, where $N_G(u)$ is the set of vertices adjacent to $u$ in $G$, and no two vertices assigned positive values are adjacent is called an textit{ independent Roman ${3}$-dominating function} (IR3DF) of $G$. The weight of an IR3DF $g$ is the sum $g(V) = sum_{v in V}g(v)$. Given a graph $G$ and a positive integer $k$, the independent Roman ${3}$-domination problem (IR3DP) is to check whether $G$ has an IR3DF of weight at most $k$. We investigate the complexity of IR3DP in bipartite and chordal graphs. The minimum independent Roman $lbrace 3 rbrace$-domination problem (MIR3DP) is to find an IR3DF of minimum weight in the input graph. We show that MIR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs. We also show that the domination problem and IR3DP are not equivalent in computational complexity aspects. Finally, we present an integer linear programming formulation for MIR3DP.