Let G be a connected graph. A non-empty subset S of V(G) is a differentiating dominating set in a graph G if it is a dominating set in G and for every two distinct vertices u and v in V(G),  A set  is a 1-movable differentiating-dominating set in G if S is a differentiating-dominating set in G and for every  either  is a differentiating-dominating set in G or there exists a vertex such that  is a differentiating-dominating set in G. The 1-movable differentiating-domination number of G, denoted by  is the smallest cardinality of a 1-movable differentiating-dominating set in G.

In this paper, the concept of 1-movable differentiating-dominating set in the join and corona of graphs are investigated. Moreover, the 1-movable differentiating-domination numbers of these graphs are determined.

differentiating domination, 1-movable domination, join, corona.