Bounds on the signed total Roman 2-domination in graphs

Abstract

Let [Formula: see text] be an integer and [Formula: see text] be a simple and finite graph with vertex set [Formula: see text]. A signed total Roman [Formula: see text]-dominating function (STR[Formula: see text]DF) on a graph [Formula: see text] is a function [Formula: see text] such that (i) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] and (ii) [Formula: see text] holds for any vertex [Formula: see text]. The weight of an STR[Formula: see text]DF [Formula: see text] is [Formula: see text] and the minimum weight of an STR[Formula: see text]DF is the signed total Roman [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, we establish some sharp bounds on the signed total Roman 2-domination number.