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 γ wcon ( GP ( n,k )) and the convex domination number γ con ( GP ( n,k )) are equal to n . For k ≥ 2 and n ≥ 13, γ wcon ( GP ( n,k )) = γ con ( GP ( n,k )) = 2 n , 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 ≥ 5, we prove that γ wcon ( J n ) = 2 n and γ con ( J n ) = 4 n .