The Signed (Total) Roman Domination Problem on some Classes of Planar Graphs -- Convex Polytopes

TL;DR

This paper investigates the exact and bounded values of signed (total) Roman domination numbers for various classes of planar graphs, providing proofs, bounds, and open problems in this graph theory area.

Contribution

It offers exact calculations for certain planar graph classes and establishes bounds for others, advancing understanding of Roman domination parameters.

Findings

Exact values for $\gamma_{sR}(A_n)$ and $\gamma_{sR}(R_n)$

Exact values for $\gamma_{stR}(S_n)$ and $\gamma_{stR}(T_n)$

Bounds for $\gamma_{sR}$ on $Q_n$, $S_n$, and $T_n$

Abstract

In this paper we deal with the calculation of the signed (total) Roman domination numbers, $\gamma_{sR}$ and $\gamma_{stR}$ respectively, on a few classes of planar graphs from the literature. We give proofs for the exact values of the numbers $\gamma_{sR}(A_n)$ and $\gamma_{sR}(R_n)$ as well as the numbers $\gamma_{stR}(S_n)$ and $\gamma_{stR}(T_n)$. For some other classes of planar graphs, such as $Q_n$, %$S_n"$ and $T_n"$, lower and upper bounds on $\gamma_{sR}$ are calculated and proved. %We give some open problems on the exact values of $\gamma_{sR}$ and $\gamma_{stR}$ for some classes of planar graphs.