Strong domination number of graphs from primary subgraphs

A set D of vertices is a strong dominating set in a graph G, if for every vertex x ∈ V (G) \ D there is a vertex y ∈ D with xy ∈ E(G) and deg(x) ≤ deg(y). The strong domination number γ_st (G) of G is the minimum cardinality of a strong dominating set in G. We consider constructions of connected graphs obtained from pairwise disjoint connected graphs by identifying or connecting some pairs of vertices. The graphs used to construct such a graph G are called the primary subgraphs of G. In this paper, we study the strong domination number of K_r -gluing of two graphs where two cliques of the same size are identified in each graph. We investigate the strong domination number for some particular cases of graphs from their primary subgraphs.