THE EDGE GEODETIC VERTEX COVERING NUMBER OF A GRAPH

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. For a connected graph G of order n≥ 2, a set S⊆V (G) is an edge geodetic vertex cover of G if S is both an edge geodetic set and a vertex covering set of G. The minimum cardinality of an edge geodetic vertex cover of G is defined as the edge geodetic vertex covering number of G and is denoted by g1α(G). Any edge geodetic vertex cover of cardinality g1α(G) is a g1α set of G. Some general properties satisfied by edge geodetic vertex cover are studied. The edge geodetic vertex covering number of several classes of graphs are determined. Connected graphs of order n with edge geodetic vertex covering number 2 is characterized. A few realization results are given for the parameter g1α(G).


INTRODUCTION
By a graph G = (V, E), we mean a finite undirected connected graph without loops and multiple edges. The order and size of G are denoted by n and m, respectively. For basic graph theoretic terminology we refer to Harary [6]. The distance d(u, v) between two vertices u and v in a connected graph G is the length of a shortest u -v path in G. A u -v path of length d (u, v) is called a u -v geodesic. It is known that this distance is a metric on the vertex set V (G). For a vertex v of G, the eccentricity e(v) is the distance between v and a vertex farthest from v. The minimum eccentricity among the vertices of G is the radius, rad G and the maximum eccentricity is its diameter, diam G. The neighborhood of a vertex v of G is the set N(v) consisting of all vertices which are adjacent with v. A vertex v is a simplicial vertex or an extreme vertex of G if the subgraph induced by its neighborhood N(v) is complete. A caterpillar is a tree of order 3 or more, the removal of whose end vertices produces a path called the spine of the caterpillar.
A geodetic set of G is a set S ⊆ V (G) such that every vertex of G is contained in a geodesic joining some pair of vertices in S. The geodetic number g(G) of G is the minimum cardinality of its geodetic sets and any geodetic set of cardinality g(G) is a minimum geodetic set or a geodetic basis or a g -set of G. The geodetic number of a graph was introduced in [2,7] and further studied in [3 -5]. A subset S ⊆ V (G) is said to be a vertex covering set of G if every edge has at least one end vertex in S. A vertex covering set of G with minimum cardinality is called a minimum vertex covering set of G. The vertex covering number of G is the cardinality of any minimum vertex covering set of G. It is denoted by α(G). The vertex covering number of a graph was studied in [10]. A set S of vertices of G is a geodetic vertex cover of G if S is both a geodetic set and a vertex covering set of G. The minimum cardinality of a geodetic vertex cover of G is defined as the geodetic vertex covering number of G and is denoted by g α (G).
Any geodetic vertex cover of cardinality g α (G) is a g α -set of G. The geodetic vertex covering number was introduced and studied in [1].
An edge geodetic cover of G is a set S ⊆ V (G) such that every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g 1 (G) of G is the minimum order of its edge geodetic covers and any edge geodetic cover of order g 1 (G) is an edge geodetic basis of G. The edge geodetic number of a graph was studied in [8]. A subset S ⊆ V (G) is a dominating set if every vertex in V − S is adjacent to at least one vertex in S. A set of vertices S in G is called an edge geodetic dominating set of G if S is both an edge geodetic set and a dominating set. The minimum cardinality of an edge geodetic dominating set of G is its edge geodetic domination number and is denoted by γ g1 (G).
The following theorems will be used in the sequel.       Throughout the following G denotes a connected graph with at least two vertices.

THE EDGE GEODETIC VERTEX COVER OF A GRAPH
Definition 2.1 Let G be a connected graph of order at least 2. A set S of vertices of G is an edge geodetic vertex cover of G if S is both an edge geodetic set and a vertex cover of G. The minimum cardinality of an edge geodetic vertex cover of G is defined as the edge geodetic vertex covering number of G and is denoted by g 1α (G). Any edge geodetic vertex cover of cardinality   Proof. Any edge geodetic set of G needs at least two vertices and so 2 ≤ max{α(G), g 1 (G)}.

Remark 2.5 The bounds in Theorem 2.4 are sharp. For the complete graph
Theorem 2.6 Each extreme vertex of G belongs to every edge geodetic vertex cover of G. In particular, each end vertex of G belongs to every edge geodetic vertex cover of G.
Proof. Since every edge geodetic vertex cover of G is also an edge geodetic set, the result follows from Theorem 1.1.

Corollary 2.7 For any graph
Proof. The result follows from Theorem 2.4 and Theorem 2.6.
Proof. Let x be the center and S = {v 1 , v 2 , ..., v n−1 } be the set of all extreme vertices of K 1,n−1 (n ≥ 3). It is clear that S is a minimum edge geodetic vertex cover of K 1,n−1 (n ≥ 3).
Proof. Since every vertex of the complete graph K n (n ≥ 2) is an extreme vertex, the vertex set of K n is the unique edge geodetic vertex cover of K n . Thus, g 1α (K n ) = n.
Now we proceed to characterize graphs G for which g 1α (G) = 2.
Then there exists at least one edge which is not incident with any of the vertices u and v. Hence S is not an edge geodetic vertex cover of Conversely, assume that there exists an edge geodetic set S = {u, v} of G such that d(u, v) ≤ 2.
It is clear that every edge of G is incident with either u or v. Hence S is a minimum edge geodetic vertex cover of G. Thus g 1α (G) = 2.

Remark 2.12
The converse of Theorem 2.11 need not be true. For the cycle , v 4 } is both a g 1 -set of C 4 and a g 1α -set of C 4 . Hence g 1 (C 4 ) = g 1α (C 4 ) = 2 but Theorem 2.14 If G is a connected graph of order n ≥ 2 with diam G ≤ 2, then g 1α (G) = g 1 (G).
Proof. Let G be a connected graph of order n ≥ 2 with diam G ≤ 2. Let S be a minimum edge geodetic set of G so that |S| = g 1 (G). If diam G = 1, then G = K n and so g 1 (G) = g 1α (G) = n.
Let diam G = 2. We show that S is a vertex cover of G. Suppose not, then there exists an edge xy ∈ E(G) such that both x, y / ∈ S. Since S is a g 1 -set of G, the edge xy lies in a u − v geodesic P : u = u 0 , u 1 , ..., u i = x, y = u i+1 , ..., u k = v for some u, v ∈ S. Then d(u, v) ≥ 3, which is a contradiction. Hence S is a minimum edge geodetic vertex cover of G and so g 1α (G) = |S| = g 1 (G).

Remark 2.15
The converse of Theorem 2.14 need not be true. For the graph G given in Figure   2.3, S = {v 1 , v 2 , v 3 , v 5 } is both a g 1 -set and a g 1α -set of G. Hence g 1α (G) = g 1 (G) = 4. But diam G = 3. Proof. Let C n : v 1 , v 2 , ..., v n , v 1 be a cycle of order n. It is clear that S = {v 1 , v 3 , v 5 , ..., v 2 n 2 −1 } is a minimum edge geodetic vertex cover of C n and so g 1α (C n ) = n 2 .
Theorem 2.17 Let G be a connected graph of order n ≥ 3. If G has exactly one vertex v of degree n − 1, then g 1α (G) = n − 1.
Proof. Let v be the unique vertex of G with degree n − 1. Then by Theorem 1.2, g 1 (G) = n − 1.
Let S = V (G) − {v} be a minimum edge geodetic set of G. It is clear that every edge of G has at least one end vertex in S. Thus S is a minimum edge geodetic vertex cover of G. Hence g 1α (G) = n − 1.

Corollary 2.18
If G has exactly one vertex v of degree n − 1, then g 1α (G) = n − 1 and G has a unique minimum edge geodetic vertex cover containing all the vertices of G other than v.

Theorem 2.21
For any tree T of order n ≥ 2, g 1α (T ) = g 1 (T ) if and only if T is a star.
Proof. Let T be a tree of order n ≥ 2. Let S be the set of all end vertices of T. Then by Theorem 1.3, S is the unique g 1 -set of T. Let g 1α (T ) = g 1 (T ). We prove that T is a star. If not, then diam T ≥ 3 and so T has at least one edge other than the end edges. Let S be the set of all edges of T which are not end edges. Then clearly no edge of S has its end vertices in S. Hence S is not a vertex cover of T. Since, by Theorem 2.6, S is a subset of any edge geodetic vertex cover of T, g 1α (T ) > |S| = g 1 (T ), which is a contradiction.

Remark 2.24
The converse of Theorem 2.23 need not be true. For the tree given in Figure 2.4, Proof. Let G be a connected graph and let S be an edge geodetic vertex cover of G. By Theorem 1.4, S is a geodetic vertex cover of G.
Theorem 2.28 For any two positive integers a and b with 3 ≤ a ≤ b, there exists a connected graph G with g α (G) = a and g 1α (G) = b. g 1α (G) = n − 1.  Proof. Let S be an edge geodetic vertex cover of G. Then S is both an edge geodetic set and a vertex cover of G. Since S is a vertex cover of G, every edge of G has at least one end in S and hence every vertex in V (G) − S has at least one neighbour in S so that S is a dominating set of Corollary 2.31 For any connected graph G, 2 ≤ γ g1 (G) ≤ g 1α (G) ≤ n.
Remark 2.32 For the graph K 2 , γ g1 (K 2 ) = 2. For the graph G given in Figure 2.2, γ g1 (G) = 2 and g 1α (G) = 3 so that γ g1 (G) < g 1α (G). For the complete graph K n , by Theorem 1.6 and Corollary 2.9, γ g1 (K n ) = g 1α (K n ) = n. Conversely, let V (G) − S be an independent set of G. Then every edge of G has at least one end in S so that S is also a vertex cover of G. Hence S is an edge geodetic vertex cover of G. Proof. We prove this theorem by considering two cases.

CONCLUSION
In this paper, we initiated the study on "The edge geodetic vertex covering number of a graph" and established some results related to this parameter. We characterized graphs G for which the edge geodetic number and the edge geodetic vertex covering number are equal. The results presented in this paper can be used for future study on the connected edge geodetic vertex covering number, the upper edge geodetic vertex covering number and forcing edge geodetic vertex covering number of a graph and so on.

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