ON HINGE DOMINATION IN GRAPHS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. A set Dh of vertices in a graph G = (V,E) is a hinge dset if every vertex u in V −Dh is adjacent to some vertex v in Dh and a vertex w in V −Dh such that (v,w) is not an edge in E. The hinge domination number γh(G) is the minimum size of a hinged dset. In this paper we determine hinge domination number γh(G) for standard graphs and some shadow distance graphs.


INTRODUCTION
A graph G = (V, E), we mean a finite, nontrivial and undirected graph without loops and multiple edges. The concept of a dset is well known in graph theoretic literature and various domination parameters have been studied. A set D h of vertices in G is called a hinge dominating set [1] if every u ∈ V − D h is adjacent to some vertex v ∈ D h and a vertex w in V − D h such that (v, w) is not an edge in E. The hinge domination number γ h (G) [1] is the minimum size of a hinge dominating set. Throughout this paper we will denote dominating set by dset.
Let D be the set of all possible distances in G = (V, E) and let D s ⊂ D. The distance graph associated with G denoted by D(G, D s ) [7] is the graph with vertex set V and two vertices u and v are adjacent in it if d(u, v) ∈ D s . The shadow distance graph of G, denoted by D sd (G, D s ) is obtained from G by considering two copies of G namely G itself and G such that if u ∈ V (G) then the corresponding vertex u is in V (G ) and E(D sd (G, D s )) = E(G) ∪ E(G ) ∪ E DS where E DS consists of the set of all edges of the form e = (u, v ) with the condition d(u, v) ∈ D s in G.
In this paper we determine the hinge domination number for some standard graphs and shadow distance graphs. We also show that the hinge domination number of the cycle graph provided in [1] is incorrect and provide the exact value.

MAIN RESULTS
We begin this section with the following result which gives the condition for a minimal hinge dset.
Thus, for at least one v ∈ D h − {u} there is a path between u and v in G. This contradicts condition (i). Also, If D h − {v} is a hinge dset, then every u ∈ V − D h is adjacent to at least one vertex in D h − {v}, so that condition (ii) also fails. Now, let us consider v ∈ D h such that v does not satisfy conditions (i) and (ii). Then from conditions (i) and (ii), D h 1 = D h − v is hinge dset. Also by condition (iii), < V − D h > is disconnected, so that D h 1 is a hinge dset of G. This contradicts condition (iii). Hence the proof.
Proof. Let D h be a minimum hinge dset of G and the number of edges t in G having one v ∈ D h and the other in then G has atleast p − q components. At least one vertex per component is requried in any hinge Theorem 2.4. For any graph G, The following result is from [1] related to the cycle graph C n .
From this result, it is clear that γ(C 3 ) = 1. As a counter example we observe that the graph figure 1 has hinge domination number 3.
We now provide the correct value of γ h (C n ) in our next result.
where computation is under modulo n.
If n = 4 and 6, the sets and for case(iii): n = 3k + 6, k = 1, 2, 3 . . . , we consider the set D h = {v 3r−2 }, 1 ≤ r ≤ n 3 . It is clear that the sets D h in cases (i), (ii) and (iii) are minimal hinge dsets. Thus, some vertex v ∈ D h is adjacent to only one vertex u ∈ V − D h and not to any other vertex. .
Hence the proof.
For the path graph P n , the following result can be found in [1].
In the next theorem, a modified version of this result is provided.
We now determine the hinge domination number for some shadow distance graphs.
when n is even and 1 ≤ k ≤ n 2 when n is odd If D h is not a hinge dset of G, there exists a vertex v ∈ D h such that D h 1 = D h − {v} is a hinge dset of G and also, < V − D h > is disconnected. This implies that D h 1 is a hinge dset of G, which contradicts condition (iii). Therefore, D h is minimal and since Hence the proof.
computation is under modulo n.
Let n ≥ 3. Then, for case(i): n = 3a, a = 1, 2, 3 . . . , we consider the set dset of G and also, < V − D h > is disconnected. This implies that D h 1 is a hinge dset of G, which contradicts condition (iii). Therefore, D h is minimal and since n ≡ 2(mod3) Hence the proof.
Therefore, D h is minimum and n − 1 n ≥ 5 Hence the proof.
Therefore, D h is minimal and Hence the proof.

CONCLUSION
In this paper, the hinge domination number of some standard graphs and shadow distance graphs related to the path and cycle graphs is determined. The hinge domination number related to the cycle C n which was provided in [1] is corrected and, a more generalized result for the hinge domination number of the path P n is provided.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.