Edge Domination in Some Path and Cycle Related Graphs

Abstract

For a graph , a subset of is called an edge dominating set of if every edge not in is adjacent to some edge in . The edge domination number of is the minimum cardinality taken over all edge dominating sets of . Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles.

1. Introduction

The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it because of its many and varied applications in such fields as linear algebra and optimization, design and analysis of communication networks, and social sciences and military surveillance. Many variants of dominating models are available in the existing literature. For a comprehensive bibliography of papers on the concept of domination, readers are referred to Hedetniemi and Laskar [1]. The present paper is focused on edge domination in graphs.

We begin with simple, finite, connected, and undirected graph of order . The set of vertices in a graph is called a dominating set if every vertex is either an element of or is adjacent to an element of . A dominating set is a minimal dominating set (or MDS) if no proper subset is a dominating set.

The minimum cardinality of a dominating set of is called the domination number of which is denoted by , and the corresponding dominating set is called a -set of .

The open neighborhood of is the set of vertices adjacent to , and the set is the closed neighborhood of .

For any real number , denotes the smallest integer not less than and denotes the greatest integer not greater than .

An edge of a graph is said to be incident with the vertex if is an end vertex of . In this case, we also say that is incident with . Two edges and which are incident with a common vertex are said to be adjacent.

In a graph , a vertex of degree one is called a pendant vertex, and an edge incident with a pendant vertex is called a pendant edge.

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 (or MEDS) 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 [2] and it is explored by many researchers. Arumugam and Velammal [3] have discussed the edge domination in graphs while the fractional edge domination in graphs is discussed in Arumugam and Jerry [4]. The complementary edge domination in graphs is studied by Kulli and Soner [5] while Jayaram [6] has studied the line dominating sets and obtained bounds for the line domination number. The bipartite graphs with equal edge domination number and maximum matching cardinality are characterized by Dutton and Klostermeyer [7] while Yannakakis and Gavril [8] have shown that edge dominating set problem is NP-complete even when restricted to planar or bipartite graphs of maximum degree . The independent edge dominating sets of certain graphs are discussed in Mojdeh and Sadeghi [9] while a constructive characterization for trees with equal edge domination and end edge domination numbers is investigated by Muddebihal and Sedamkar [10]. The edge domination in graphs of cubes is studied by Zelinka [11].

Throughout the paper, and will denote the path and the cycle with vertices, respectively.

We will give brief summary of definitions which are useful for the present investigations.

Definition 1. The open neighborhood of an edge is denoted as and it is the set of all edges adjacent to in . Further, is the closed neighborhood of in .

Definition 2. 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 3 (see [12]). The line graph of , written , is the simple graph whose vertices are the edges of , with when and have a common end vertex in .

Definition 4 (see [13]). The shadow graph of a connected graph is constructed by taking two copies of , say, and . Join each vertex of to the neighbors of the corresponding vertex of . The shadow graph of is denoted by .

Definition 5 (see [13]). The middle graph of a connected graph denoted by is the graph whose vertex set is where two vertices are adjacent if(i)they are adjacent edges of , or(ii)one is a vertex of and the other is an edge incident with it.

Definition 6 (see [14]). The total graph of denoted by is the graph whose vertex set is and two vertices are adjacent in if(i)they are adjacent edges of , or(ii)one is a vertex of and the other is an edge incident with it, or(iii)they are adjacent vertices of .
It is easy to see that always contains both and the line graph as its induced subgraph. The total graph is the largest graph that is formed by the adjacency relations of elements of a graph.
For the various graph theoretic notations and terminology, we follow West [12] while the terms related to the concept of domination are used in the sense of Haynes et al. [15].
Generally, the following types of problems are considered in the field of domination in graphs:(1)to introduce new types of dominating models,(2)to determine bounds in terms of various graph parameters,(3)to obtain the exact domination number for some graphs or graph families,(4)to study the algorithmic and complexity results for particular dominating parameters, and(5)to characterize the graphs with certain dominating parameters.
The present work is intended to discuss the problem of the third kind in the context of edge domination in graphs. In this paper, we investigate the edge domination number of middle graphs, total graphs, and shadow graphs of and .

2. Main Results

Theorem 7. .

Proof. Consider two copies of . Let be the vertices of the first copy of and let be the vertices of the second copy of . Let be the edges of the first copy of and let be the edges of the second copy of where connects and and connects and . Then and .
For , is obviously an MEDS with minimum cardinality among all minimal edge dominating sets of . Hence, .
For , we construct an edge set of as follows: where with .
Since each edge in is either in or is adjacent to an edge in , it follows that the above set is an edge dominating set of .
Moreover, the above set is an MEDS of because for any edge , the set does not dominate the edges in of . Now, for and and pairs of edges for will dominate maximum number of distinct edges of . Therefore, any set containing the edges less than that of cannot be an edge dominating set of . This implies that the above edge dominating set is of minimum cardinality.
Hence, the above set is an MEDS with minimum cardinality among all minimal edge dominating sets of .
Thus, .

Theorem 8. .

Proof. Let be the vertices of path and let be the added vertices corresponding to the edges of to obtain . Thus, .
Then and . Let the edges where .
Now, the edge sets and are clearly the minimal edge dominating sets of and , respectively, with minimum cardinality among all minimal edge dominating sets of . Hence, for .
For , we construct an edge set of as follows:
Since each edge in is either in or is adjacent to an edge in , it follows that the above set is an edge dominating set of . Moreover, the above set is an MEDS of because for any edge , the set does not dominate the edges in of .
Now, each graph , for , has two nonadjacent pendant edges and there is no edge which is adjacent to both pendant edges. Hence, at least two distinct edges are required to dominate these pendant edges. Moreover, for and for ; each edge of at most distinct edges out of total edges of can dominate seven distinct edges of including itself and each of the remaining edges can dominate less than six distinct edges of at a time. Therefore, any set containing edges less than that of cannot be an edge dominating set of . This implies that the above edge dominating set is of minimum cardinality.
Hence, the above set is an MEDS with minimum cardinality among all minimal edge dominating sets of .
Thus, .

Theorem 9. For path ,

Proof. Let be the vertices of path and let be the added vertices corresponding to the edges of to obtain . Thus, . The graph will have vertices and edges. Let the edges where for odd and for even .
Now, we construct the edge sets of as follows: for with , and if , for with .
The above set is an edge dominating set of because each edge in is either in or is adjacent to an edge in . Also, since for any edge , the set does not dominate the edges in of , it follows that the above set is an MEDS of .
Now, the sets and are clearly minimal edge dominating sets of and respectively with minimum cardinality. Hence, and . For , implying that an edge of can dominate at most seven distinct edges of including itself. But, from the nature of graph, we can observe that each of at most distinct edges of can dominate seven distinct edges including itself and each of the remaining edges can dominate less than six edges of . Therefore, any set containing the edges less than that of cannot be an edge dominating set of . This implies that the above edge dominating set is of minimum cardinality.
Hence, the above set is an MEDS with minimum cardinality among all minimal edge dominating sets of .
This implies that

Theorem 10. For cycle ,

Proof. Consider two copies of . Let be the vertices of the first copy of and let be the vertices of the second copy of . Let be the edges of the first copy of and be the edges of the second copy of . Then and .
First, we construct an edge set of as follows: with for or and for .
Since each edge in is either in or is adjacent to an edge in , it follows that the above set is an edge dominating set of .
Now, the above set is an MEDS of because for any edge , the set does not dominate the edges in of . Moreover, for and each edge of can dominate at most seven distinct edges of including itself. But, at a time, each of at most distinct edges of can dominate seven distinct edges of including itself and each of the remaining edges can dominate less than six distinct edges of . Therefore, any set containing the edges less than that of cannot be an edge dominating set of . This implies that the above edge dominating set is of minimum cardinality.
Hence, the above set is an MEDS with minimum cardinality among all minimal edge dominating sets of .
Thus,

Theorem 11. .

Proof. Let be the vertices of cycle and let be the added vertices corresponding to the edges of to obtain . Then and . Let the edges , where .
Now, we construct an edge set of as follows:
Since each edge in is either in or is adjacent to an edge in , it follows that the above set is an edge dominating set of . Moreover, the above set is an MEDS of because for any edge , the set does not dominate the edges in of .
Now, for and each edge of can dominate at most seven distinct edges of including itself. But, at a time, each of at most edges of can dominate seven distinct edges of including itself and each of the remaining edges can dominate less than six distinct edges of . Therefore, any set containing the edges less than that of cannot be an edge dominating set of . This implies that the above edge dominating set is of minimum cardinality.
Hence, the above set is an MEDS with minimum cardinality among all minimal edge dominating sets of .
Thus, .

Theorem 12. For cycle

Proof. Let be the vertices of cycle and let be the added vertices corresponding to the edges of to obtain . Then and . Let the edges where for odd and for even .
First, we construct an edge set of as follows: for , with if and if .
The above set is an edge dominating set of because each edge in is either in or is adjacent to an edge in . Since for any edge , the set does not dominate the edges in of , it follows that the above set is an MEDS of .
Now, for and each edge of can dominate at most seven distinct edges of including itself. But, at a time, each of at most distinct edges of can dominate seven distinct edges of including itself and each of the remaining edges can dominate less than six distinct edges of . Therefore, any set containing the edges less than that of cannot be an edge dominating set of . This implies that the above edge dominating set is of minimum cardinality.
Hence, the above set is an MEDS with minimum cardinality among all minimal edge dominating sets of .
Thus,

3. Concluding Remarks

Here, we have taken up a problem to determine the edge domination number for the larger graphs obtained by means of three graph operations on paths and cycles. To derive similar results in the context of other variants of domination is an open area of research.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are highly thankful to the anonymous referees for their kind comments and fruitful suggestions on the first draft of this paper.