Restrained Edge Domination Number of Some Path Related Graphs

For a graph , a set is a restrained dominating (restrained edge dominating) set if every vertex (edge) not in S is adjacent (incident) to a vertex (edge) in S and to a vertex (edge) in The minimum cardinality of a restrained dominating (restrained edge dominating) set of G is called restrained domination (restrained edge domination) number of G, denoted by The restrained edge domination number of some standard graphs are already investigated while in this paper the restrained edge domination number like degree splitting, switching, square and middle graph obtained from path.


Introduction
The theory of domination in graphs has received considerable attention due to its diversified applications to handle many real life situations such as design analysis of network to military surveillance and linear algebra to social sciences.
The graphs considered here are simple, finite, connected, undirected with vertex set and edge set . The minimum degree among the vertices of graph is denoted by while the maximum degree among the vertices of graph is denoted by . The open neighborhood of is the set . An edge of a graph is said to be incident with the vertex if is an end vertex of A set is a dominating set if every vertex of is either in or is adjacent to atleast one element of The minimum cardinality of a dominating set of is called domination number and it is denoted by The edge analogue of dominating set is also available. A subset is an edge dominating set if each edge in is either in or is adjacent to an edge in An edge dominating set is called a minimal edge dominating set if no proper subset of is an edge dominating set. The edge domination number is the minimum cardinality among all minimal edge dominating sets. The concept of edge domination was introduced by Mitchell and Hedetniemi [1]. Vaidya and Pandit [2] have discussed edge domination in some path and cycle related graphs and edge domination in graphs is explored by Arumugam and Velammal [3].
For graph theoretic terminology and notation we rely on Harary [4]. A brief account of dominating set and its related concepts can be found in Haynes et al. [5]. Many variants of dominating sets are available in the existing literature. One such variant is restrained domination, a set is a restrained dominating set if every vertex not in is adjacent to a vertex in as well as to a vertex in The minimum cardinality of a restrained dominating set is called the restrained domination number of , denoted by The concept of restrained domination was introduced by Telle and Proskurowski [6] as a vertex partitioning problem. Restrained domination in the context of path and cycle is discussed by Vaidya and Ajani [7,8] while the restrained domination of complete graph, multipartite graphs and the graphs with minimum degree two is well studied by Domke et al. [9,10]. Moreover the concept of total equitable bondage number was introduced by Vaidya and Parmar [11,12] while equi independent equitable domination was explored by Vaidya and Kothari [13]. These variants are introduced by identifying one or more characteristics of elements of vertex subset or edge subset.
The present work is focused on edge analogue of restrained domination in graphs. For a graph , a set is a restrained edge dominating set if every edge not in is incident to an edge in and also incident to an edge in The minimum cardinality of restrained edge dominating set of is called restrained edge domination number, denoted as . This concept was introduced by Soner and Ghobadi [14] and further explored by Vaidya and Ajani [15].
The restrained edge domination numbers are investigated for the graphs obtained from degree splitting, switching of a vertex, square and middle graph of path. Definition 1.1: The degree of an edge of is defined by and it is equal to the number of edges adjacent to it. The maximum degree of an edge in is denoted by Definition 1.2: Let be a graph with , where each is a set of all the vertices of same degree with at least two elements and ⋃ . The degree splitting graph is obtained from G by adding vertices and joining to each vertex of for . Note that if ⋃ then Definition 1.3: The switching of a vertex v of G means removing all the edges incident to v and adding edges joining v to every vertex which is not adjacent to v in G. We denote the resultant graph by ̃. Definition 1.4: The square of a graph G denoted by has the same vertex set as of G and two vertices are adjacent in if they are at distance of 1 or 2 apart in G.

Theorem 2.1:
If is the graph obtained by degree splitting of then, ( ) ⌈ ⌉ Proof: Let be the vertices and be the edges of Let and be the set of all the vertices of same degree with at least two elements. In order to obtain from path We add two vertices and corresponding to and such that each vertex of and joined by an edge to and respectively. Let  for . So the edges in above set will dominate maximum number of distinct edges of Therefore any set containing the edges less than the number of edges in set will not dominate all the edges of This implies that above set is the restrained edge dominating set with minimum cardinality.
Hence set is a minimal restrained edge dominating set with minimum cardinality among all the minimal restrained edge dominating set of Thus, ( ) ⌈ ⌉.

Illustration 2.2:
The degree splitting of is shown in Fig. 1 where the set of edges is its restrained edge dominating set of minimum cardinality. with | | ⌈ ⌉. Note that ̃ . As each edge in is adjacent to an edge in and to another edge in , it follows that is a restrained edge dominating set of ̃. For any edge the set does not dominate the edges in of ̃, which implies that the above set is a minimal restrained edge dominating set of ̃. Now ( ̃) for and also for . Due to adjacency nature, the edges in above set will dominate maximum number of distinct edges of ̃. Therefore any set containing the edges less than the number of edges in the set will not dominate all the edges of ̃. This implies that above set is the restrained edge dominating set with minimum cardinality.
Hence set is a minimal restrained edge dominating set with minimum cardinality among all the minimal restrained edge dominating set of ̃ Thus ( ̃) ⌈ ⌉, for .

Illustration 2.4:
The switching of path is shown in Fig. 2 where the set of edges is its restrained edge dominating set of minimum cardinality. Fig. 2. ( ̃) .

Theorem 2.5:
If is the graph obtained from square of path then ⌈ ⌉. with | | ⌈ ⌉. Since and each edge in is adjacent to an edge in and to another edge in It follows that the set is a restrained edge dominating set of . Moreover the above set is a minimal restrained edge dominating set of because for any edge the set does not dominate the edges in of Now for and So the edges in the set will dominate maximum number of distinct edges of . Therefore any set containing the edges less than the number of edges in the set will not dominate all the edges of . This implies that above set is the restrained edge dominating set with minimum cardinality.
Hence set is a minimal restrained edge dominating set with minimum cardinality among all the minimal restrained edge dominating set of Thus, ⌈ ⌉ Illustration 2.6: The square of path is shown in Fig. 3 where the set of edges is its restrained edge dominating set of minimum cardinality.   Fig. 4 where the set of edges is its restrained edge dominating set of minimum cardinality. .