Edge Domination in Web Graph

Let γγee(GG) be the edge domination number of a graph. A “web graph” WW(ss, tt) is obtained from the Cartesian product of cycle graph of order ss and path graph of order tt. In this paper, edge domination number of the web graph is determined. Mathematical subject classification: 05C69


Introduction
Let = ( , ) be a graph with its vertex set = ( ) and edge set = ( ). In a graph an edge ∈ is said to be incident with vertex ∈ if the vertex is an end vertex of the edge . Two edges of a graph are adjacent if they are distinct and have a common vertex. The open neighborhood of an edge ∈ ( ) denotes as (e) is the set of all edges that adjacent to while the closed neighborhood of e is the set [e] = ( ) ∪ { }. We use | | to denote the cardinality of a set D.
Mitchell and Hedetniemi [6] introduced the definition of "edge domination in the graphs".
A subset of the edge set in graph is called an edge dominating set of if every edge not in is adjacent to some edge in . The "edge domination ‫اﻟﻌﺎﺷر‬ ‫اﻟدوﻟﻲ‬ ‫اﻟﻌﻠﻣﻲ‬ ‫اﻟﻣؤﺗﻣر‬ ‫واﺳط‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺗرﺑﯾﺔ‬ ‫ﻛﻠﯾﺔ‬ number" of a graph which we denote by ( ) is the minimum cardinality taken over all edge dominating sets of . In this work, edge domination of the "web graph" is calculated. Haynes et.al [4] gave the treatment of the "Fundamentals of Domination" in graph theory. For more details on several advanced topics about domination in graphs see [1-2, 5, 7-11]. Any term or notation about the graph in this paper can be found in [3].

Definition 1.3. A "web graph"
( , ) is obtained from the "Cartesian product" of cycle graph of order and path graph of order .

Web graph
In this work we are describing the "web graph" in terms of cycles and paths. In this graph there are copies of cycles C as shown by dashed lines and copies of paths as shown by bold lines in Figure 1 as an example. In general, we describe these cycles and paths as follows.

‫اﻟﻌﺎﺷر‬ ‫اﻟدوﻟﻲ‬ ‫اﻟﻌﻠﻣﻲ‬ ‫اﻟﻣؤﺗﻣر‬ ‫واﺳط‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺗرﺑﯾﺔ‬ ‫ﻛﻠﯾﺔ‬
We labeled the vertices of this graph by where represents the cycle label starting from the inside out, and is the vertex label of of j ℎ cycle ordering clockwise. While represents the path label ordering clockwise, and is the label of vertex in ℎ path starting from the inside out. For example, see Figure1. Proof. To get the minimum number of edges so that we can dominate all edges of the graph, we must search the edges which have maximum closed neighborhood. There are three types of edges that depend on the cardinal number of closed neighborhood as follows.
if the edge ( ) belongs to the 1 or of the graph 6, if the edge ( ) belongs to the terminal edges of any path 7, otherwise � Thus, we get the minimum dominating set when we choose the edges from the paths by the following method.

‫اﻟﻌﺎﺷر‬ ‫اﻟدوﻟﻲ‬ ‫اﻟﻌﻠﻣﻲ‬ ‫اﻟﻣؤﺗﻣر‬ ‫واﺳط‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺗرﺑﯾﺔ‬ ‫ﻛﻠﯾﺔ‬
We choose and let edges, from in alternating way, for is odd. By this choice, all edges of these paths as well as all edges of the cycles are dominated (as an example, see bold edges in Figure 2).
In the same manner in Theorem 2.1, the set 1 ∪ 2 dominates all edges of when ≢ 1 ( 3)(as an example, see bold edges in Figure 3). Otherwise there are two cases as follows.  The edges of are not adjacent to any edges of 1 and 2 , so we can dominate the last cycle by � 3 � edges by using Remark 1.1(i), (as an example, see bold edges in Figure 4). Thus, we get the result. Case 3. ≢ 1 ( 3), then in the same manner in the previous case the set 1 dominates all edges of , for is odd and all edges in every cycles , = 1, … , except one edge ( ) in last cycle .
2 dominates the edges of , for is even. So, the set 1 ∪ 2 ∪ { } dominates all edges of (with minimum cardinality) in this case (as an example, see bold edges in Figure 5). Therefore we get the result.