Degree equitable restrained double domination in graphs

A subsetD ⊆ V (G) is called an equitable dominating set of a graphG if every vertex v ∈ V (G)\D has a neighbor u ∈ D such that |dG(u) − dG(v)| ≤ 1. An equitable dominating set D is a degree equitable restrained double dominating set (DERD-dominating set) of G if every vertex of G is dominated by at least two vertices of D, and 〈V (G) \ D〉 has no isolated vertices. The DERDdomination number of G, denoted by γ cl(G), is the minimum cardinality of a DERD-dominating set of G. We initiate the study of DERD-domination in graphs and we obtain some sharp bounds. Finally, we show that the decision problem for determining γ cl(G) is NP-complete.


Introduction
Let G = (V, E) be a graph. The number of vertices of G we denote by n and the number of edges we denote by m, thus |V (G)| = n and |E(G)| = m. The complement of G, denoted byḠ, is a graph which has the same vertices as If u ∈ pn[u, S] and u is an isolated vertex in S , then u is called its own private neighbor. By a leaf we mean a vertex of degree one, while a support vertex is a vertex adjacent to a leaf. We say that a support vertex is weak if it is adjacent to exactly one leaf. We say that a vertex is isolated if it has no neighbor. Let ∆(G) mean the maximum degree among all vertices of G. The path (cycle, respectively) on n vertices we denote by P n (C n , respectively). A wheel W n , where n ≥ 4, is a graph with n vertices, formed by connecting a vertex to all vertices of a cycle C n−1 . The distance between two vertices of a graph is the number of edges in a shortest path connecting them. The eccentricity of a vertex is the greatest distance between it and any other vertex. The diameter of a graph G, denoted by diam(G), is the maximum eccentricity among all vertices of G. By K p,q we denote a complete bipartite graph with partite sets of cardinalities p and q. By a star we mean the graph K 1,q . By a double star we mean a graph obtained from a star by joining a positive number of vertices to one of its leaves. Generally, let K t 1 ,t 2 ,...,t k denote the complete multipartite graph with vertex set The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. For a comprehensive survey of domination in graphs, see [4,5].
A subset D ⊆ V (G) is a restrained dominating set of G if every vertex of V (G) \ D has a neighbor in D as well as a neighbor in V (G) \ D. The restrained domination number of G, denoted by γ r (G), is the minimum cardinality of a restrained dominating set of G. A restrained dominating set of G of minimum cardinality is called a γ r (G)-set.
A dominating set D of a graph G is said to be a cototal dominating set of G if the induced subgraph V (G) \ D has no isolated vertices. The cototal domination number of G, denoted by γ cl (G), is the minimum cardinality of a cototal dominating set of G. Restrained domination in graphs was introduced by Domke et. al [1]. Independently, Kulli et. al [9] initiated the study of cototal domination in graphs. The concepts of restrained domination and cototal domination are equivalent.
A subset D ⊆ V (G) is a double dominating set of G if every vertex of G is dominated by at least two vertices of D. The double domination number of G, denoted by γ d (G), is the minimum cardinality of a double dominating set of G. The study of double domination in graphs was initiated by Harary and Haynes [3].
A subset D ⊆ V (G) is a restrained double dominating set of G if every vertex of G is dominated by at least two vertices of D, and no vertex of V (G) \ D is isolated. The restrained double domination number of G, denoted by γ dcl (G), is the minimum cardinality of a restrained double dominating set of G. The study of restrained double domination in graphs was initiated by in [8].
The equitable domination number of G, denoted by γ e (G), is the minimum cardinality of an equitable dominating set of G. The concept of equitable domination in graphs was introduced by V. Swaminathan and K. Dharmalingam [11] by considering the following real world situation. In a network, nodes with nearly equal capacity may interact with each other in a better way. In societies, persons with nearly equal statuses tend to be friendly. For more details on the domination refer [6,7,10,12].
We introduce a new variant of equitable domination, namely the degree equitable restrained double domination (DERD-domination), and we initiate the study of this parameter. An equitable dominating set D of a graph G is said to be a DERD-dominating set of G if every vertex of G is dominated by at least two vertices of D, and V (G) \ D has no isolated vertices. The DERDdomination number of G, denoted by γ e cl (G), is the minimum cardinality of a DERD-dominating set of G.

Results
Since the one-vertex graph, as well as all graphs with an isolated vertex, does not have a DERD-dominating set, in this paper we consider only graphs without isolated vertices.
We begin with the following straightforward observations. Observation 1. Let G be a graph without isolated vertices. Then every DERD-dominating set of G contains all leaves and support vertices of G.

Observation 2.
There is no graph G such that γ e cl (G) = n − 1.  We have the following property of regular and (k, k + 1)-biregular graphs.
Proof. Let D be a minimum restrained double dominating set of G. Let u ∈ V (G) \ D. Thus there exist vertices w, v ∈ D such that uw, uv ∈ E(G). We have |d . This implies that γ e cl (G) = γ dcl (G).
Theorem 9. For every graph G we have 2 ≤ γ e cl (G) ≤ n. Further, the lower bound is attained if and only if G = K 2 or G = K n − {x} where x is any vertex in K n ; n ≥ 5 and the upper bound is attained if and only if G does not contain an edge uv ∈ E(G) which satisfies the following conditions: Proof. Lower bound follows from the definition of DERD-set. Now consider the equality of lower bound. Suppose γ e cl (G) = 2 and G = K n or K n − {x}. Then G contains at least two vertices u, v ∈ V (G) such that {u, v} contains no edge. Let D be DERD-set of G such that u, v / ∈ D. Let w, x ∈ D. Since u and v are independent vertices in G, therefore w and x must be adjacent to both u and v also by the definition of DERD-set V − D contains no isolated vertices. Therefore, we need at least one more vertex to compliance the necessary conditions required to define DERD-set in G. Hence |D| ≥ 3, a contradiction.
Conversely, suppose G = K n , then by Observation 3, γ e cl (G) = 2 and if G = K n − {x}; n ≥ 5, then any two adjacent vertices will form a DERD-set for G. Proof. Let T be a tree and γ e cl (T ) = n. Suppose T does not satisfies the hypothesis of the theorem, then there exist at least an edge uv ∈ E(T ) incident to exactly four support vertices x, y, z, w such that N (x) ∩ N (y) = {u} and N (z) ∩ N (w) = {v} which implies that V − {u, v} is isomorphic to K 2 . Therefore |D| = n − 2. Hence γ e cl (T ) = |D| = n − 2, a contradiction. Conversely, suppose G does not contain an edge uv ∈ E(T ) as stated in the hypothesis of the theorem, then V − D = π, which implies that |D| = n. Hence γ e cl (T ) = |D| = n.
By Observation 2, there exists no graph with γ e cl (T ) = n − 1. We now consider trees T such that γ e cl (T ) ≤ n − 2.
Let S(n, k)-star (where n ≥ 2 and k ≥ 1) be a tree obtained from a path P n making each vertex v i ∈ V (P n ) (2 ≤ i ≤ n) adjacent to least k new leaves. We have |V (S(n, k))| = n + k and |E(S(n, k))| = n + k − 1.
Operation O: Let v be a support vertex of a tree T . Attach |d G (v) − 1| or |d G (v) − 2| leaves to at least one leaf adjacent to v, and attach exactly one leaf to other leaves adjacent to v.
Let T be the family of trees such that T = {T : T is obtained from a star by a finite sequence of operations O}.
We now characterize the trees with γ e cl (T ) = n − 2.
Theorem 11. If T is a tree with at least six vertices, then γ e cl (T ) = n − 2 if and only if T ∈ T and T is obtained from a S(2, k)-star (k ≥ 2) by a finite sequence of operations O.
Similarly, we can characterize the trees with γ e cl (T ) = k (k ≥ 3) by S(n, n − k)-star by finite sequence of operations O.
We need the following theorem to prove our further results. 3. Complexity issues for γ e cl (G) To show that the DERD-domination decision problem for arbitrary graphs is NP-complete, we shall use a well known NP-completeness result called Exact Three Cover (X3C), which is defined as follows.
EXACT COVER BY 3-SETS (X3C). Instance: A finite set X with |X| = 3m and a collection C of 3-element subsets of X. Question: Does C contain an exact cover for X, that is, a subcollection C ⊆ C such that every element of X occurs in exactly one member of C ? Note that if C exists, then its cardinality is precisely m.

DEGREE EQUITABLE RESTRAINED DOUBLE DOMINATING SET (DERDdominating set).
Instance: A graph G = (V, E) and a positive integer k ≤ |V |. Question: Is there a DERD-dominating set of cardinality at most k?
Theorem 15. DERD-dominating set problem is NP-complete, even for bipartite graphs.
Proof. It is clear that the DERD-dominating set problem is NP. To show that it is NP-complete, we establish a polynomial transformation from X3C. Let X = {x 1 , x 2 , . . . , x 3m } and C = {c 1 , c 2 , . . . , c m } be an arbitrary instance of X3C. We construct a bipartite graph G and a positive integer k such that this instance of X3C will have an exact 3-cover if and only if G has a DERD-dominating set of cardinality at most k. With each edge x i ∈ X, associate a path P 4 with vertices x i , y i , z i , t i , with each c j associate a path P 3 with vertices c j , d j , s j . Then add new vertices u 1 , u 2 , . . . , u 2m , and make them adjacent to all x j s. The construction of a bipartite graph G is completed by joining x i and c j if and only if x i ∈ c j . Finally, set k = 2m + 9m.