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 < 4 then both the weakly convex domination number wcon(GP(n, k)) and the convex domination number con(GP(n, k)) are equal to n. For k < 2 and n < 13, wcon(GP(n, k)) = 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 Jn, where n is odd and n < 5, we prove that wcon(Jn) = 2n and con(Jn) = 4n.