The signed (total) Roman domination problem on some classes of planar graphs — Convex polytopes

Abstract

In last two decades, the basic Roman domination problem and the variants thereof have attracted many researchers from different fields to study these problems in both ways, theoretically and computationally. Function [Formula: see text] is the signed Roman domination function (SRDF) if and only if it satisfies the following conditions: (i) for each vertex [Formula: see text] of graph [Formula: see text] the sum of the values assigned to [Formula: see text] and all its neighbors is at least 1 and (ii) each vertex [Formula: see text] for which [Formula: see text], must be adjacent to at least one vertex [Formula: see text] such that [Formula: see text]. The minimum weight [Formula: see text] of all SRDFs on graph [Formula: see text] is defined as signed Roman domination number ([Formula: see text]). Definition of the signed total Roman domination function (STRDF) follows the definition of SRDF with a minor change where condition (i) considers the sum of all values assigned to all neighbors of [Formula: see text] to be at least one, for each vertex [Formula: see text]. The minimum weight [Formula: see text] of all STRDFs on graph [Formula: see text] is called signed total Roman domination number ([Formula: see text]). In this paper, we deal with the calculation of the signed (total) Roman domination numbers on a few classes of planar graphs from the literature. We give proofs for the exact values of the numbers [Formula: see text] and [Formula: see text] as well as the numbers [Formula: see text] and [Formula: see text]. For some other classes of planar graphs, such as [Formula: see text], and [Formula: see text], lower and upper bounds on [Formula: see text] are presented.