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

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.