Weakly convex and convex domination numbers for generalized Petersen and flower snark graphs

Abstract

We consider the weakly convex and convex domination numbers for two classes of graphs: generalized Petersen graphs and flower snark graphs. For a given generalized Petersen graph $GP(n,k)$, we prove that if $k=1$ and $n\geq 4$ then both the weakly convex domination number $\gamma_\mathit{wcon}(GP(n,k))$ and the convex domination number $\gamma_\mathit{con}(GP(n,k))$ are equal to $n$. For $k\geq 2$ and $n\geq 13$, $\gamma_\mathit{wcon}(GP(n,k))=\gamma_\mathit{con}(GP(n,k))=2n$, which is the order of $GP(n,k)$. Special cases for smaller graphs are solved by the exact method. For a flower snark graph $J_n$, where $n$ is odd and $n\geq 5$, we prove that $\gamma_\mathit{wcon}(J_n)=2n$ and $\gamma_\mathit{con}(J_n)=4n$.