Gallai-type theorems and domination parameters Original Research Article

Let γ(G) denote the minimum cardinality of a dominating set of a graph G = (V,E). A longstanding upper bound for γ(G) is attributed to Berge: For any graph G with n vertices and maximum degree Δ(G), γ(G) ⩽ n − Δ(G). We characterise connected bipartite graphs which achieve this upper bound. For an arbitrary graph G we furnish two conditions which are necessary if γ(G) + Δ(G) = n and are sufficient to achieve n − 1 ⩽ γ(G) + Δ(G) ⩽ n. We further investigate graphs which satisfy similar equations for the independent domination number, i(G), and the irredundance number ir(G). After showing that i(G) ⩽ n − Δ(G) for all graphs, we characterise bipartite graphs which achieve equality. Lastly, we show for the upper irredundance number, IR(G): For a graph G with n vertices and minimum degree δ(G), IR(G) ⩽ n - δ(G). Characterisations are given for classes of graphs which achieve this upper bound for the upper irredundance, upper domination and independence numbers of a graph.