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.
Highlights
Let G = (V, E) be a connected undirected graph without loops and parallel edges
A vertex set X ⊆ V is a weakly convex set in G if for every two vertices u, v ∈ X, there exists at least one shortest u − v path, whose vertices belong to X
We prove that in the case of the generalized Petersen graph GP (n, k), for k ≥ 2 and n ≥ 13, both minimal weakly convex dominating set and minimal convex dominating sets must contain all vertices
Summary
A vertex set X ⊆ V is a weakly convex set in G if for every two vertices u, v ∈ X, there exists at least one shortest u − v path (in G), whose vertices belong to X. A vertex set X ⊆ V is a convex set if for every two vertices u and v from X, every shortest u − v path (in G) belongs to X . If two u vertices ui, uj belong to a weakly convex dominating set S, at least one of the sets {ui+1, ui+2, .
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