More results on the signed double Roman domination number of graphs

Abstract

A signed double Roman dominating function (SDRD-function) on a graph G is defined as a function [Formula: see text] having the property that [Formula: see text] for each [Formula: see text] and if [Formula: see text], then the vertex u must have a neighbor w with [Formula: see text] or two neighbors assigned 2 under f, and if [Formula: see text], then v must have at least one neighbor w with [Formula: see text]. The weight of an SDRD-function f is the value [Formula: see text]. The signed double Roman domination number[Formula: see text] is the minimum weight of an SDRD-function. It is conjectured that the signed double Roman domination number of a nontrivial graph G is bounded above by its order. In this paper we prove this conjecture for cactus graphs, and we present some new bounds on [Formula: see text]. We also determine the signed double Roman domination number of perfect binary trees.

Keywords

• Roman domination
• signed double Roman domination
• trees
• 05C69