The restrained double Roman domination and graph operations

Abstract

A restrained double Roman dominating function (RDRD-function) on a graph G is a function $f:V\left(G\right)\to \{0,1,2,3\}$ that satisfies two conditions: (1) If $f\left(v\right)<2$, then ${\sum }_{u\in N_G\left[v\right]}f\left(u\right)\ge \left|A{N}_G^f\right(v\left)\right|+2$, where $A{N}_G^f\left(v\right)=\{u\in N_G(v):f(u)\ge 1\}$; (2) The subgraph induced by the vertices assigned 0 under f contains no isolated vertices. The weight of an RDRD-function f is ${\sum }_{v\in V\left(G\right)}f\left(v\right)$, and the minimum weight of an RDRD-function on G is defined as the restrained double Roman domination number (RDRD-number) of G, denoted by ${\gamma }_{rdR}\left(G\right)$. In this paper, we first establish that computing the RDRD-number is NP-hard, even for chordal graphs. Then the impact of various graph operations, including the strong product, cardinal product, and corona product, on the restrained double Roman domination number are given.