Convex and Weakly Convex Domination in Prism Graphs

For a given graph G = (V;E) and permutation π : V ↦ V the prism πG of G is defined as follows: V (πG) = V (G) ∪ V (G′), where G′ is a copy of G, and E(πG) = E(G) ∪ E(G′) ∪ M_π, where M_π = {uv′ : u ∈ V (G); v = π (u)} and v′ denotes the copy of v in G′. We study and compare the properties of convex and weakly convex dominating sets in prism graphs. In particular, we characterize prism γcon-fixers and -doublers. We also show that the differences γ_wcon(G) – γ_wcon(πG) and γ_wcon (πG) – 2γ_wcon (G) can be arbitrarily large, and that the convex domination number of πG cannot be bounded in terms of γ_con (G).