eu 1-MOVABLE CLIQUE DOMINATING SETS OF A 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 mcl(G) (resp. γ 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 mcl(G) is called a γ 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. AMS Subject Classification: 05C69

A nonempty subset S of V (G) is a dominating set of G if for every v ∈ V (G) \ S, there exists u ∈ S such that uv ∈ E(G), that is N G [S] = V (G).The domination number of G, denoted by γ(G), is the minimum cardinality among all dominating sets of G.A dominating set S of G with |S| = γ(G) is called a γ-set of G. Domination in graph as well as its variants and the numerous applications of these related concepts in networks have been widely studied.The book by Haynes et al. (see [6]) contains a long list of some variations of the standard domination concept.Now, given two vertices u and v of G, by I G [u, v] we mean the closed interval consisting of u, v and all vertices lying on some u − v geodesic (shortest path connecting u and v) of G.A subset C of V (G) is convex if I G (u, v) ⊆ C for every pair of vertices u, v ∈ C. A proper convex subset of V (G) of largest cardinality is called a maximum convex set in G.The cardinality of a maximum convex set in G is called the convexity number of G and is denoted by con(G).
The concept of clique domination is studied by Cozzens and Kelleher in [4].Convexity and some of its related concepts are studied and investigated in [2], [3], and [5] while convex domination in graphs are dealt with in [8] and [9].Blair et al. introduced and studied movable domination in [1].The concept of 1-movable domination is also studied in [7].
This paper investigates, among others, the concept of 1-movable clique domination in the join and composition of graphs.Recall that the join of two graphs G and H is the graph

Results
The first result shows that the concepts of 1-movable clique domination and 1-movable convex domination are equivalent.Proof.Suppose that S is a 1-movable convex dominating set of G. Suppose further that S is not complete.Then there exist vertices x, y ∈ S such that d(x, y) = 2. Let z ∈ S ∩ N G (x) ∩ G (y). Then S \ {z} and (S \ {z}) ∪ {v} are not convex sets for all v ∈ (V (G) \ S) ∩ N G (z).This implies that S is not a 1-movable convex dominating set of G, contrary to our assumption.Thus, S is a clique dominating set of G. Next, let u ∈ S. Since S is a 1-movable convex dominating set of G, either S {v} is a convex dominating set or there exists a vertex w ∈ (V (G) S) ∩ N G (u) such that S u = (S {u}) ∪ {w} is a convex dominating set of G.If S {u} is a convex dominating set, then it S {v} is a clique dominating set since S \ {u} is a clique (because S is a clique).Suppose (S {v}) ∪ {w} is a convex dominating set of G for some w ∈ (V (G) S)∩N G (u). Suppose further that there exists q ∈ (S \{u})\N G (w). Then [q, u, w] is a q-w geodesic.Hence, (S {v}) ∪ {w} is not a convex set, contrary to our assumption.Therefore S {v}) ∪ {w} is a clique dominating set of G. Accordingly, S is a 1-movable clique dominating set of G.
The converse follows from the fact that every subset of V (G) that induces a clique is a convex set.
The next result is a direct consequence of Theorem 2.1.
Theorem 2.4.Let G be a connected nontrivial graph.Then the following are equivalent: mcl (G) = 2 if and only if there exist adjacent vertices x and y of G satisfying the following properties: Proof.Let G be a connected graph of order n ≥ 3 such that γ 1 mcl (G) = 2. Let S = {x, y} be a γ 1 mcl -set of G. Then x and y are adjacent vertices.Clearly, . This proves (iii).Similarly, (iv) holds.
For the converse, suppose that there exist x, y ∈ V (G) such that x and y are adjacent and satisfy conditions (i) to (iv).We claim that S = {x, y} is a 1-movable clique dominating set of G. Clearly, S = K 2 , and so is complete.Let a ∈ V (G) S. Then by (i), a ∈ (N G (x) ∪ N G (y)) {x, y}.This means that a = x, a = y and ax ∈ E(G) or ay ∈ E(G).Since a is arbitrary, S is a dominating set of G. Consequently, S is a clique dominating set of G. Now consider {x}.If {x} is a dominating set of G, then γ(G) = 1.By Theorem 2.4, . Hence, pv ∈ E(G).Consequently, {x, v} is a clique dominating set of G. Similarly, if we consider y, then either {y} is a clique dominating set of G or there exists w ∈ (V (G) S)∩N G (x) such that {y, w} is a clique dominating set of G.This shows that S is a 1-movable clique dominating set of G. Suppose (iii) S = {a, b}, where one of the following is satisfied: (1) a and b are not isolated vertices of G and H, respectively.
(2) {a} is a dominating set of G.
(3) {a, v} is a clique dominating set of G for some v ∈ V (G).
(4) {b} is a dominating set of H.
(5) {b, w} is a clique dominating set of H for some w ∈ V (H). (iv Since S is a 1-movable clique dominating set of G + H, a and b cannot be both isolated vertices.If a and b are both non-isolated vertices, then (1) of (iii) holds.Suppose that a or b is an isolated vertex, say b is an isolated vertex of 3) of (iii) holds.Similarly, (4) or ( 5) of (iii) holds if a is an isolated vertex.Subcase 2.
Let S 1 = {a}.Since S is a clique, S 2 is a clique in H.If a is not an isolated vertex of G, then we are done.Suppose a is an isolated vertex of G. Since S is a 1-movable clique dominating set G + H, either S {a} = S 2 is a clique dominating set of H or there exists v ∈ V (H) S 2 such that (S {a}) ∪ {v} = S 2 ∪ {v} is a clique dominating set of H. Hence, (iv) holds.Similarly, (v) holds if Since S is a clique and S = S 1 ∪ S 2 , it follows that S 1 and S 2 are cliques in G and H, respectively.
The converse is easy.
The next result is an immediate consequence of Theorem 2.6.
Corollary 2.7.Let G and H be nonempty graphs.Then 1 ≤ γ 1 mcl (G + H) ≤ 2.Moreover, γ 1 mcl (G+H) = 1 if and only if one of the following statements holds: We now characterize the 1-movable clique dominating sets in the composition of graphs.
This means that (T x {a}) ∪ {b} = {b} is a clique dominating set of H. Thus, T x is a 1-movable clique dominating set of H.If x = y, then y ∈ (V (G) S) ∩ N G (x).Hence, (S {x}) ∪ {y} = {y} is a clique dominating set of G. Thus, S is a 1-movable clique dominating set of G showing that (iii) holds.
For the converse, suppose that S is a clique dominating set of G and (i), (ii) and (iii) hold.Then, clearly, Let G = (V (G), E(G)) be a graph with n = |V (G)| and m = |E(G)|.For any vertex v ∈ V (G), we define the open neighborhood of v as the set N G (v) = {u ∈ V (G) : uv ∈ E(G)} and the closed neighborhood of v as the set N G [v] = N G (v) ∪ {v}.If S is a nonempty subset of X, then N G (S) = v∈S N G (v) and N G [S] = N G (S) ∪ S.

Theorem 2 . 8 .
Let G and H be connected nontrivial graphs such that G has a clique dominating set.A subset C = x∈S [{x} × T x ], where S ⊆ V (G) and T x ⊆ V (H) for each x ∈ S, is a 1-movable clique dominating set of G[H] if and only if S is a clique dominating set of G such that (i) T x is a clique in H for each x ∈ S (ii) T x is a clique dominating set of H whenever S = {x} and |T x | ≥ 2. (iii) T x is a 1-movable clique dominating set of H or T x is a clique dominating set of H and S is a 1-movable clique dominating set of G whenever S = {x} and |T x | = 1.Proof.Suppose that C is a 1-movable clique dominating set of G[H].Then C is a clique dominating set of G[H].Hence, S is a clique dominating set of G. Let x ∈ S and let a, b ∈ T x , where a
where |E| ≥ 2, a is not an isolated vertex of G and E is a clique in H, or E is a clique dominating set of H, or E ∪ {v} is a clique dominating set of H for some v ∈ V (H) E. (v) S = D ∪ {b}, where |D| ≥ 2, b is not an isolated vertex of H and D is a clique in G, or D is a clique dominating set of G, or D ∪ {w} is a clique dominating set of G for some w ∈ V (G) D. (vi) |S 1 | ≥ 2 and |S 2 | ≥ 2, where S 1 and S 2 are cliques in G and H, respectively.
Proof.Suppose S is a 1-movable clique dominating set of G + H. Consider the following cases: Case