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 γwcon(GP(n,k)) and the convex domination number γcon(GP(n,k)) are equal to n. For k≥2 and n≥13, γwcon(GP(n,k))=γcon(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 γwcon(Jn)=2n and γcon(Jn)=4n.