Bounds on the strong domination number Original Research Article

Let G=(V(G),E(G)) a graph. A set D⊆V(G) is a strong dominating set of G, if for every vertex x∈V(G)−D there is a vertex y∈D with xy∈E(G) and d(x,G)⩽d(y,G). The strong domination number γst(G) is defined as the minimum cardinality of a strong dominating set and was introduced by Sampathkumar and Pushpa Latha (Discrete Math. 161 (1996) 235–242). In this paper we present some sharp upper bounds on γst(G) depending on the existence of certain cycles in G. The sharpness of our results is established by characterizing all graphs which achieve the given upper bound under our assumptions on the cycles. Furthermore, we pose a conjecture about the influence of the minimum degree δ(G) on γst(G).