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.