The Restrained Double Roman Domination in Graphs

Abstract

A double Roman dominating function on a graph $G=(V(G),E(G))$ is a function $f: V(G) \rightarrow \{0,1,2,3\}$ satisfying the property that every vertex assigned 0 has at least two neighbors assigned 2 or one neighbor assigned 3, and every vertex assigned 1 has at least one neighbor assigned 2 or 3. A double Roman dominating function f is called a restrained double Roman dominating function (RDRD-function) if the induced subgraph of G by the vertices assigned 0 under f has no isolated vertex. The weight of an RDRD-function f is the value $w(f)=\sum _{v \in V(G)} f(v)$, and the minimum weight over all RDRD-functions on G is the restrained double Roman domination number (RDRD-number) $\gamma _{\textrm{rdR}}(G)$ of G. In this paper, we first characterize the graphs with small RDRD-numbers and then show the sharp bounds of $\gamma _{\textrm{rdR}}(G)+\gamma _{\textrm{rdR}}(\overline{G})$ for any connected graph G with order at least 3. Finally, a linear time algorithm for computing the RDRD-number of any cograph is presented. These results partially answer two open problems posed by Mojdeh et al. (Rairo-Oper Res 4(56):2293–2304 (2022). https://doi.org/10.1051/ro/2022089).