Edge Detour Monophonic Number of a Graph

For a connected graph G of order at least two, an edge detour monophonic set of G is a set S of vertices such that every edge of G lies on a detour monophonic path joining some pair of vertices in S. The edge detour monophonic number of G is the minimum cardinality of its edge detour monophonic sets and is denoted by edm(G). We determine bounds for it and characterize graphs which realize these bounds. Also, certain general properties satisfied by an edge detour monophonic set are studied. It is shown that for positive integers a, b and c with 2 ≤ a ≤ b ≤ c, there exists a connected graph G such that m(G) = a,m1(G) = b and edm(G) = c, where m(G) is the monophonic number and m1(G) is the edge monophonic number of G. Also, for any integers a and b with 2 ≤ a ≤ b, there exists a connected graph G such that dm(G) = a and edm(G) = b, where dm(G) is the detour monophonic number of a graph G.


Introduction
By a graph G = (V, E) we mean a finite undirected connected graph without loops or multiple edges.The order and size of G are denoted by n and m, respectively.For basic graph theoretic terminology we refer to Harary [4].For vertices x and y in a connected graph G, the distance d(x, y) is the length of a shortest x −y path in G.An x−y path of length d(x, y) is called an x−y geodesic.The neighborhood of a vertex v is the set N (v) consisting of all vertices u which are adjacent with v.A vertex v is an extreme vertex if the subgraph induced by its neighbors is complete.A vertex v in G is said to be a semi-extreme vertex of G if ∆ (< N(v) >) = |N (v)| − 1.That is, the induced subgraph of N (v) has a full degree vertex in N(v).
For the graph G given in Figure 1.1, v 2 , v 3 , v 4 , v 5 and v 6 are the semiextreme vertices.In any graph G, each extreme vertex is a semi-extreme vertex.
The closed interval I[x, y] consists of all vertices lying on some x − y geodesic of G, while for S ⊆ V, I[S] = S x,y∈S I[x, y].A set S of vertices is a geodetic set if I[S] = V, and the minimum cardinality of a geodetic set is the geodetic number g(G).A geodetic set of cardinality g(G) is called a g-set.The geodetic number of a graph was introduced in [1,5] and further studied in [2].
A chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called monophonic if it is a chordless path.A set S of vertices of a graph G is a monophonic set if each vertex v of G lies on an x − y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G, denoted by m(G).A longest x−y monophonic path is called an x−y detour monophonic path.A set S of vertices of a graph G is a detour monophonic set if each vertex v of G lies on an x − y detour monophonic path for some x, y ∈ S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G).The detour monophonic number of a graph was introduced in [9] and further studied in [10].
An edge monophonic set of G is a set S of vertices such that every edge of G lies on a monophonic path joining some pair of vertices in S.
The edge monophonic number of G is the minimum cardinality of its edge monophonic sets and is denoted by m 1 (G).An edge monophonic set of cardinality m 1 (G) is an m 1 -set of G.
These concepts have interesting applications in Channel Assignment Problem in radio technologies, and the detour matrix of a connected graph is used to discuss the applications of the detour index and hyper-detour index to a class of graphs, which in turn, capture different aspects of certain molecular graphs associated to the molecules arising in special situations of molecular problems in theoretical Chemistry [3,6].Also, there are useful applications of these concepts to security based communication network design.In the case of designing the channel for a communication network, although all the vertices are covered by the network when considering detour monophonic sets, some of the edges may be left out.This drawback is rectified in the case of edge detour monophonic sets so that considering edge detour monophonic sets is more advantageous to real life application of communication networks.This motivated us to introduce and investigate edge detour monophonic sets in a graph.
The following theorems will be used in the sequel.
Theorem 1.1.[8] Each extreme vertex of a graph G belongs to every monophonic set of G.
Theorem 1.2.[7] Each semi-extreme vertex of a graph G belongs to every edge monophonic set of G.
Theorem 1.3.[9] Each extreme vertex of a graph G belongs to every detour monophonic set of G.
Throughout this paper G denotes a connected graph with at least two vertices.

Edge detour monophonic number of a graph
Definition 2.1.Let G be a connected graph with at least two vertices.An edge detour monophonic set of G is a set S of vertices such that every edge of G lies on a detour monophonic path joining some pair of vertices in S. The edge detour monophonic number of G is the minimum cardinality of its edge detour monophonic sets and is denoted by edm(G).An edge detour monophonic set of cardinality edm(G) is an edm-set of G.
We observe that every edge detour monophonic set is also a detour monophonic set of G.
Example 2.2.For the graph G given in Figure 2.1, it is easily seen that no 3-element subset of vertices is an edge detour monophonic set.It is clear that S 1 = {z, v, w, x} is an edge detour monophonic set of G so that edm(G) = 4. Also, S 2 = {z, v, w, u}, S 3 = {z, x, u, v} and S 4 = {z, x, u, w} are minimum edge detour monophonic sets of G.
Note that for the graph G given in Figure 2.1, it is easily verified that {z, v} is a minimum monophonic set of G and {z, v, w} is a minimum detour monophonic set of G so that m(G) = 2 and dm(G) = 3.Thus the monophonic number, detour monophonic number and edge detour monophonic number of a graph are different. Proof.
An edge detour monophonic set needs at least two vertices and so edm(G) ≥ 2. Clearly, the set of all vertices of G is an edge detour monophonic set of G so that edm(G) ≤ n. 2 The bounds in Theorem 2.3 are sharp.The even cycle C n (n ≥ 4) has edm(C n ) = 2 and the complete graph K n has edm(K n ) = n.Theorem 2.4.Each semi-extreme vertex of a graph G belongs to every edge detour monophonic set of G.In particular, if the set S of all semiextreme vertices of G is an edge detour monophonic set, then S is the unique minimum edge detour monophonic set of G.

Proof.
Let S be the set of all semi-extreme vertices of G and let T be any edge detour monophonic set of G. Suppose that there exists a vertex Since T is an edge detour monophonic set of G, the edge e = uv lies on an x − y detour monophonic path P : which is a contradiction to the fact that P is an x−y detour monophonic path.Hence S is contained in every edge detour monophonic set of G. 2 Corollary 2.5.For any graph G with k semi-extreme vertices, max{2, k} ≤ edm(G) ≤ n.
Corollary 2.6.For the complete graph

Proof.
Suppose that there is a component B of G − v such that B contains no vertex of S. Let u be any vertex in B and let e be any edge incident with u, say e = uw.Since S is an edge detour monophonic set, there exist vertices x, y ∈ S such that e lies on some x−y detour monophonic path P : x = u 0 , u 1 , . . ., u, w, . . ., u t = y in G with u 6 = x, y.Let P 1 be the x − u subpath of P and P 2 be the u − y subpath of P .Since v is a cutvertex of G, both P 1 and P 2 contain v, so that P is not a path, which is a contradiction.Thus every component of G − v contains an element of S. 2 Theorem 2.9.For any connected graph G, no cut-vertex of G belongs to any minimum edge detour monophonic set of G.
Proof.Let v be a cut-vertex of G and let S be a minimum edge detour monophonic set of G. Then by Theorem 2.8, every component of G − v contains an element of S. Let U and W be two components of G − v and let u ∈ U and w ∈ W . Then v is an internal vertex of any u − w detour monophonic path.Let S 0 =S − {v}.It is clear that every edge that lies on an u − v detour monophonic path also lies on an u − w detour monophonic path.Hence it follows that S 0 is an edge detour monophonic set of G, which is a contradiction to S a minimum edge detour monophonic set of G. 2 Theorem 2.10.If T is a tree with k end vertices, then edm(T ) = k.
Proof.This follows from Theorems 2.4 and 2.9. 2 Theorem 2.11.For the cycle C n (n ≥ 3), edm (Cn) = ( } is a minimum edge detour monophonic set of C n and so edm(C n ) = 2.If n is odd, then clearly S = {v 1 , v 2 , v 3 } is a minimum edge detour monophonic set of C n and so edm(C n ) = 3. 2 Theorem 2.12.For the complete bipartite graph Proof.(i) This follows from Theorem 2.10.(ii) Let r ≥ 2 and let U = {u 1 , u 2 , . . ., u r } and W = {w 1 , w 2 , . . ., w s } be a bipartition of G. Let S = U .We prove that S is an edm-set of G.We observe that any u − v detour monophonic path in G is of length at most 2. Any edge u i w j (1 ≤ i ≤ r, 1 ≤ j ≤ s) lies on the detour monophonic path u i w j u k for any k 6 = i so that S is an edge detour monophonic set of G. Let T be any set of vertices such that |T | < |S|.If T 6 U , then there exists a vertex u i ∈ U such that u i / ∈ T .Then any edge u i w j (1 ≤ j ≤ s), does not lie on a detour monophonic path joining a pair of vertices of T .Thus T is not an edge detour monophonic set of G.If T 6 W , then the argument is similar.If T 6 S ∪ W such that T contains at least one vertex from each of S and W , then since |T | < |S|, there exist vertices u i ∈ U and w j ∈ W such that u i / ∈ T and w j / ∈ T .It is clear that the edge u i w j does not lie on a detour monophonic path joining any pair of vertices of T so that T is not an edge detour monophonic set of G. Hence S is an edge detour monophonic set with minimum cardinality so that edm(G) = |S| = r. 2 A vertex v in a graph G is called an independent vertex if the subgraph induced by its neighbours contains no edges.Theorem 2.13.Let G be a connected graph.Then edm(G) = 2 if and only if there exist two independent vertices u and v such that every edge of G lies on a u − v detour monophonic path.

Proof.
Let edm(G) = 2 and let S = {u, v} be an edge detour monophonic set of G.If u and v are adjacent, then the graph is G = K 2 , and the result is true.Suppose that u and v are non-adjacent in G.We prove that u and v are independent vertices.Suppose that u is not an independent vertex.Then there exists an edge xy such that x, y ∈ N (u).It is clear that the edge xy does not lie on any u − v detour monophonic path so that S is not an edge detour monophonic set, which is a contradiction.The converse is trivial. 2 Theorem 2.14.Let G be a connected graph of order n.If G has more than one vertex of degree n − 1, then every edge detour monophonic set contains all vertices of degree n − 1.
Proof.Let G be a graph of order n with more than one vertex of degree n − 1.If u and v are two vertices of degree n − 1, then uv is an edge and it is not an edge of any detour monophonic path joining two vertices of G other than u and v. Hence it follows that both u and v belong to every edge detour monophonic set of G. 2 Theorem 2.15.For any graph G of order n with at least two vertices of degree n − 1, edm(G) = n.

Proof.
If all the vertices are of degree n − 1, then G = K n and so edm(G) = n.Otherwise, let v 1 , v 2 , . . ., v k (2 ≤ k ≤ n − 2) be the vertices of degree n − 1. Suppose that edm(G) < n.Let S be a edm-set of G such that |S| < n.By Theorem 2.14, S contains all the vertices v 1 , v 2 , . . ., v k .Let v be a vertex such that v / ∈ S. Then deg(v) < n − 1.Since any two of v 1 , v 2 , . . ., v k are adjacent, the edge vv i (1 ≤ i ≤ k) does not lie on a detour monophonic path joining a pair of vertices v j and v l (j 6 = l).Similarly, since any v j is adjacent to any vertex of S, which is different from v 1 , v 2 , . . ., v k , the edge vv i (1 ≤ i ≤ k) does not lie on a detour monophonic path joining a vertex v j and a vertex of S, which is different from v 1 , v 2 , . . ., v k .Now, let u and w be two vertices of S different from v 1 , v 2 , . . ., v k .Since v i is adjacent to both u and w, the edge vv i does not lie on a detour monophonic path joining u and w.Thus we see that the edges vv i (1 ≤ i ≤ k) do not lie on any detour monophonic path joining a pair of vertices of S, which is a contradiction to S an edge deour monophonic set of G. Hence edm(G) = n. 2 Remark 2.16.The converse of Theorem 2.15 is not true.For the graph G given in Figure 1.1, S = {v 1 , v 2 , v 3 , v 4 , v 5 , v 6 } is a minimum edge detour monophonic set of G so that edm(G) = 6 = n and has exactly one vertex v 1 of degree n − 1.
Theorem 2.17.Let G be a graph of order n ≥ 3.If G contains a cutvertex of degree n − 1, then edm(G) = n − 1.
Proof.Let v be a cut-vertex of degree n − 1. Clearly S = V − {v} is an edge detour monophonic set of G and so edm(G) ≤ n − 1.Now, we show that edm(G) = n − 1.Let T be any set of vertices with |T | ≤ n − 2. Then there exist at least two vertices, say u and w, which are not in T .Since v is adjacent to all the remaining vertices of G, the edges vu and vw do not lie on any detour monophonic path joining any two vertices of T. Hence T is not an edge detour monophonic set of G and so edm(G) = n − 1. 2 Remark 2.18.The converse of Theorem 2.17 is not true.For the graph G given in Figure 2.3, S = V (G) − {y} is an edm-set and so edm(G) = 4.However, y is a cut-vertex of degree 3.

Realisation Results
Theorem 3.1.For every pair k, n of integers with 2 ≤ k ≤ n, there exists a connected graph G of order n with edm(G) = k.

Proof.
For k = n, let G = K n .Then by Corollary 2.6, we have edm(G) = n.Now, let 2 ≤ k < n.Let G be any tree of order n with k end vertices.Then by Theorem 2.10, edm(G) = k. 2

Remark 2 . 7 .
The graph G given in Figure2.2 is non-complete on 4 vertices with edm(G) = 4. Theorem 2.8.Let G be a connected graph with cut-vertices and S an edge detour monophonic set of G.If v is a cut-vertex of G, then every component of G − v contains an element of S.

Theorem 3 . 2 .
For any integers a, b and c with 2 ≤ a ≤ b ≤ c, there exists a connected graph G such that m(G) = a, m 1 (G) = b and edm(G) = c.Proof.We consider four cases.Case 1.For a = b = c, any tree with a end vertices has the desired property.Case 2. a < b = c.Subcase (i). a = b−1.Let C : v 1 , v 2 , . . ., v 6 , v 1 be a cycle of order 6.Let G be the graph obtained by adding a new vertices u 1 , u 2 , . . ., u a to C and joining each u i (1 ≤ i ≤ a−1) to v 1 , joining u a to v 4 , and joining the vertices v 3 and v 5 .The graph G is shown in Figure 3.1.Let S = {u 1 , u 2 , . . ., u a } be the set of all extreme vertices of G.By Theorems 1.1, 1.2 and 2.4, S is a subset of every monophonic set, edge monophonic set and edge detour monophonic set of G. Clearly, S is a monophonic set and so m(G) = a.It is easily seen that S is not an edge monophonic set and an edge detour monophonic set of G, since the edge v 3 v 5 does not lie any x −y monophonic set of G and so edm(G) = c.Subcase (ii).a ≤ b − 2. Let P 3 : x, y, z be a path of order 3, let P 5 : v 1 , v 2 , . . ., v 5 be a path of order 5 and let P b−a+1 : u 1 , u 2 , . . ., u b−a+1 be a path of order b−a+1.Let H be the graph obtained from P 3 , P 5 and P b−a+1 by joining x to v 2 ; z to v 4 ; v 4 to u b−a+1 ; and each vertex u i (1 ≤ i ≤ b−a+1) to v 5 .Let G be the graph obtained by adding c + a − b − 2 new vertices z 1 , z 2 , . . ., z a−1 , w 1 , w 2 , . . ., w c−b−1 to H and joining each z i (1 ≤ i ≤ a − 1) to v 1 , and joining each w i (1 ≤ i ≤ c − b − 1) to both the vertices v 2 , v 4 .The graph G is shown in Figure 3.5.Let S = {z 1 , z 2 , . . ., z a−1 , u 1 } be the set of all extreme vertices of G.