Let G be a nontrivial connected graph. A nonempty subset S of V (G) is a clique dominating set of G if S is a dominating set and the induced subgraph S of S is complete. The minimum cardinality among all clique dominating sets of G, denoted by γ cl (G), is called the clique domination number of G. A clique dominating set S of G with |S| = γ cl (G) is called a γ cl-set of G. This study aims to characterize the clique dominating sets in the join, corona, composition and cartesian product of graphs and determine the corresponding clique domination number of the resulting graph.

A clique (convex) dominating set S of G is a 1-movable clique dominating set (resp. 1-movable convex dominating set) of G if for every v ∈ S, either S \ {v} is a clique (resp. convex) dominating set or there exists a vertex u ∈ (V (G) \ S) ∩ NG(v) such that (S \ {v}) ∪ {u} is a clique (resp. convex) dominating set of G. The minimum cardinality of a 1-movable clique (resp. 1-movable convex) dominating set of G, denoted by γ 1 mcl (G) (resp. γ 1 mcon (G)), is called the 1-movable clique domination number (resp. 1-movable convex domination number) of G. A 1-movable clique dominating set in G with cardinality γ 1 mcl (G) is called a γ 1 mcl -set of G. This paper aims to characterize the 1-movable clique dominating sets of some graphs including those resulting from the join and composition of two graphs. The corresponding 1-movable clique domination number of the resulting graph is then determined. Further, it is shown that the concepts of 1-movable clique domination and 1-movable convex domination are equivalent.