Connected Edge Monophonic Number of a Graph

For a connected graph G of order n, a set S of vertices is called an edge monophonic set of G if every edge of G lies on a monophonic path joining some pair of vertices in S, and the edge monophonic number m e (G) is the minimum cardinality of an edge monophonic set. An edge monophonic set S of G is a connected edge mono-phonic set if the subgraph induced by S is connected, and the connected edge monophonic number m ce (G) is the minimum cardinality of a connected edge monophonic set of G. Graphs of order n with connected edge monophonic number 2, 3 or n are characterized. It is proved that there is no non-complete graph G of order n ≥ 3 with m e (G) = 3 and m ce (G) = 3. It is shown that for integers k, l and n with 4 ≤ k ≤ l ≤ n, there exists a connected graph G of order n such that m e (G) = k and m ce (G) = l. Also, for integers j, k and l with 4 ≤ j ≤ k ≤ l, there exists a connected graph G such that m e (G) = j, m ce (G) = k and g ce (G) = l, where g ce (G) is the connected edge geodetic 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 [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.It is known that the distance is a metric on the vertex set of G.An x-y path of length d(x, y) is called an x-y geodesic.A vertex v is said to lie on an x-y geodesic P if v is a vertex of P including the vertices x and y.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 of G if the subgraph induced by its neighbors is complete.
A vertex v is a semi-extreme vertex of G if the subgraph induced by its neighbors has a full degree vertex in N(v).In particular, every extreme vertex is a semi-extreme vertex and a semi-extreme vertex need not an extreme vertex.For the graph G in Figure 2.1, v 1 and v 3 are an extreme vertices as well as semi-extreme vertices.Also v 2 is a semi-extreme vertex and not an extreme vertex of G.
A set S of vertices is a geodetic set of G if every vertex of G lies on a geodesic joining some pair of vertices in S, 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 of G.The geodetic number of a graph was introduced in [1,5] and further studied in [2,3,6].It was shown in [5] that determining the geodetic number of a graph is an NP-hard problem.A set S of vertices in G is called an edge geodetic set of G if every edge of G lies on a geodesic joining some pair of vertices in S, and the minimum cardinality of an edge geodetic set is the edge geodetic number g e (G) of G.An edge geodetic set of cardinality g e (G) is called a g e -set of G.An edge geodetic set S of G is called a connected edge geodetic set of G if the subgraph induced by S is connected, and the minimum cardinality of a connected edge geodetic set is the connected edge geodetic number g ce (G) of G.A connected edge geodetic set of cardinality g ce (G) is called a g ce -set of G.The edge geodetic number of a graph was introduced and studied in [8,9].
A chord of a path u 1 , u 2 , . . ., u k in G is an edge u i u j with j ≥ i + 2. A u-v path P is called a monophonic path if it is a chordless path.A set S of vertices is a monophonic set if every vertex of G lies on a monophonic path joining some pair of vertices in S, and the minimum cardinality of a monophonic set is the monophonic number m(G) of G.A monophonic set of cardinality m(G) is called an m-set of G.A set S of vertices in G is called an edge monophonic set of G if every edge of G lies on a monophonic path joining some pair of vertices in S, and the minimum cardinality of an edge monophonic set is the edge monophonic number m e (G) of G.An edge monophonic set of cardinality m e (G) is called an m e -set of G.The edge monophonic number of a graph was introduced and studied in [7].
Theorem 1.1.[7] Every semi-extreme vertex of a connected graph G belongs to each edge monophonic set of G.In particular, if the set S of all semi-extreme vertices of G is an edge monophonic set of G, then S is the unique minimum edge monophonic set of G.
Theorem 1.2.[7] Let G be a connected graph with cut-vertices and S an edge monophonic set of G.If v is a cut-vertex of G, then every component of G − v contains an element of S.
(2) For any non-trivial tree T of order n with k endvertices, m e (T ) = k.

Connected edge monophonic number of a graph
Definition 2.1.Let G be a connected graph with at least two vertices.A connected edge monophonic set of G is an edge monophonic set S such that the subgraph induced by S is connected.The minimum cardinality of a connected edge monophonic set of G is the connected edge monophonic number of G and is denoted by m ce (G).A connected edge monophonic set of cardinality m ce (G) is called an m ce -set of G.
Example 2.2.For the graph G given in Figure 2.1, it is easily seen that no 4-element subset of vertices is an edge monophonic set.It is clear that It is easily seen that no 6-element subset of vertices is a connected edge monophonic set of G.
Note that S 1 = {v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 10 } is also a minimum connected edge monophonic set of G. Thus the edge monophonic number and connected edge monophonic number are different.Proof.An edge monophonic set needs at least two vertices and so m e (G) ≥ 2. Since every connected edge monophonic set is also an edge monophonic set, it follows that m e (G) ≤ m ce (G).Also, since the set of all vertices of G is a connected edge monophonic set of G, m ce (G) ≤ n. 2 We observe that for the complete graph K 2 , m ce (K 2 ) = m e (K 2 ) = 2 and for the complete graph K n (n ≥ 3), m ce (G) = m e (G) = n.Also, all the inequalities in Theorem 2.3 are strict.For the graph G given in Figure 2.1, m e (G) = 5, m ce (G) = 7 and n = 10.Theorem 2.4.Every semi-extreme vertex of a connected graph G belongs to each connected edge monophonic set of G.In particular, if the set S of all semi-extreme vertices of G is a connected edge monophonic set of G, then S is the unique minimum connected edge monophonic set of G.
Proof.Since every connected edge monophoic set is also an edge monophonic set, the result follows from Theorem 1.1.2 Corollary 2.5.For any connected graph G of order n with k semi-extreme vertices, max{2, k} ≤ m ce (G) ≤ n.
Proof.This follows from Theorems 2.3 and 2.4. 2 Corollary 2.6.For the complete graph The converse of Corollary 2.6 need not be true.For the graph G given in Figure 2.2, each vertex is a semi-extreme and S = {v 1 , v 2 , v 3 , v 4 , v 5 , v 6 } is an m ce -set of G. Therefore, m ce (G) = 6 and G is not a complete graph.
Since every connected edge monophonic set is an edge monophonic set, the next theorem follows from Theorem 1.2.Proof.Let G be a connected graph and S a connected edge monophonic set of G. Let v be a cut-vertex of G and G 1 , G 2 , . . ., G r (r ≥ 2) the components of G − v.By Theorem 2.7, S contains at least one vertex from each Combining Theorems 2.4 and 2.8, we have the following theorem.
Theorem 2.9.Every semi-extreme vertex and every cut-vertex of a connected graph G belong to each connected edge monophonic set of G.
Since every connected edge geodetic set is a connected edge monophonic set, we have the following theorem.
Theorem 2.10.Every semi-extreme vertex and every cut-vertex of a connected graph G belong to each connected edge geodetic set of G.
Corollary 2.11.For any connected graph G of order n with k semiextreme vertices and l cut-vertices, max{2, k Proof.This follows from Theorems 2.3 and 2.9 2 Corollary 2.12.For any tree T of order n, m ce (T ) = n.
Proof.This follows from Corollary 2.11. 2 Theorem 2.13.For the complete bipartite graph We prove that S is a minimum connected edge monophonic set of G.Note that any u -v monophonic path in G is of length at most 2. Every edge u i w j (1 ≤ i ≤ r, 1 ≤ j ≤ s) lies on the monophonic path u i , w j , u k for any k 6 = i, and so S is a connected edge monophonic set of G. Let T be any set of vertices such that If T is such that T contains vertices from U and W such that u i / ∈ T and w j / ∈ T .Then clearly the edge u i w j does not lie on a monophonic path joining two vertices of T so that T is not a connected edge monophonic set.Thus in any case T is not a connected edge monophonic set of G. Hence S is a minimum connected edge monophonic set so that m ce (G) = |S| = r +1. 2 Theorem 2.14.For any cycle Proof.
Let S = {u, v} be a minimum connected edge monophonic set of G. Then uv is an edge.If G 6 = K 2 , then there exists an edge xy different from uv, and the edge xy does not lie on any u-v monophonic path so that S is not a connected edge monophonic set, which is a contradiction.Thus G = K 2 . 2 A vertex v in graph G is called an independent vertex if the subgraph induced by its neighbors contains no edges.
Theorem 2.16.Let G be a non-complete connected graph of order n ≥ 3. Then m ce (G) = 3 if and only if there exist two independent vertices u and w such that d(u, w) = 2 and every edge of G lies on a u-w monophonic path.
Proof.Let m ce (G) = 3 and let S = {u, v, w} be a connected edge monophonic set of G.If the subgraph induced by S is complete, then G ∼ = K 3 , which is a contradiction.So assume that u and w are non-adjacent in G.It is clear that d(u, w) = 2. Now, we show that u and w are independent vertices of G. Suppose that u is not an independent vertex of G. Then there exist vertices u 1 , u 2 ∈ N (u) such that u 1 and u 2 are adjacent in G. Since S is a connected edge monophonic set, u 1 u 2 lies on a u-w monophonic path P .Since u 1 and u 2 are adjacent to u in G, it follows that P is not a monophonic path, which is a contradiction.Thus, u is an independent vertex of G. Similarly, w is an independent vertex of G. Now, since S is a connected edge monophonic set and since the subgraph induced by S is the path P : u, v, w, it follows that every edge of G lies on a u-w monophonic path.Conversely, let u and w be two independent vertices of G such that d(u, w) = 2 and every edge of G lies on a u-w monophonic path.Since G is non-complete, no 2-element subset of G is a connected edge monophonic set of G. Now, let P : u, v, w be a u-w monophonic path.Then S = {u, v, w} is a minimum connected edge monophonic set of G so that m ce (G) = 3. 2 Corollary 2.17.If G is a non-complete graph of order n ≥ 3 with m ce (G) = 3, then m e (G) = 2. Proof.This follows from Theorems 2.20 and 2.9. 2 The converse of Theorem 2.24 is not true.For the graph G given in Figure 2.4, S = {v 1 , v 3 , v 4 , v 5 , v 6 } is the set of all semi-extreme vertices of G and v 2 is the only cut-vertex of G. Hence by Theorem 2.9, m ce (G) = 6 = n.However, G has no cut-vertex of degree n − 1.
Theorem 2.25.Let G be a connected graph of order n.If G has at least two vertices of degree n−1, then every vertex of G is a semi-extreme vertex of G.
Proof.Let u 1 , u 2 , . . ., u l (l ≥ 2) be the vertices of degree n − 1.Then each u i (1 ≤ i ≤ l) is adjacent to all other vertices in G. Let u be any vertex in G.The converse of Theorem 2.26 is not true.For the graph G given in Figure 2.2, S = {v 1 , v 2 , v 3 , v 4 , v 5 , v 6 } is a minimum connected edge monophonic set of G so that m ce (G) = 6 = n and G has no vertex of degree n − 1.
Theorem 2.27.Let G be a connected graph of order n.Then m ce (G) = n if and only if every vertex of G is either a cut-vertex or a semi-extreme vertex.
Proof.Let m ce (G) = n.Then S = V is the only connected edge monophonic set of G. Suppose that there exists a vertex u such that u is neither a semi-extreme vertex nor a cut-vertex of G. Since u is not a semi-extreme vertex, for each v ∈ N (u), there exists a vertex w ∈ N (u) such that w 6 = v and vw is not an edge of G. Now, we show that S = V − {u} is an edge monophonic set of G. Let vu be any edge of G. Then vu lies on the monophonic path P : v, u, w.Also, any edge xy not incident with u lies on the x-y monophonic path itself.Hence S is an edge monophonic set of G. Since u is not a cut-vertex of G, the subgraph induced by S is connected.Therefore, S is a connected edge monophonic set of G so that m ce (G) ≤ n − 1, which is a contradiction.Hence every vertex of G is either a semi-extreme vertex or a cut-vertex.The converse follows from Theorem 2.9. 2 Corollary 2.28.For the graph G = K 1 + ∪m j K j , where

Realisation Results
In view of Corollary 2.18, we have the following realization result.Let S 1 = {u 3 , u 4 , . . ., u l−1 , u l } be the set of semi-extreme vertices and cutvertices of G.Then, for any vertex y / ∈ S 1 , S 1 ∪ {y} is not a connected edge monophonic set of G.It is clear that S 1 ∪ {u 1 , u 2 } is a connected edge monophonic set of G so that, by Theorem 2.9, m ce (G) = l.If l = 3, then {u 1 , u 3 } is an edge monophonic set of G so that m e (G) = 2 = k.It is clear that no 2-element subset of vertices is a connected edge monophonic set of G. Since {u 1 , u 2 , u 3 } is a connected edge monophonic set, it follows that m ce (G) = 3 = l.Then for any vertex y / ∈ S 1 , S 1 ∪ {y} is not a connected edge monophonic set of G.It is clear that S 1 ∪ {u 1 , u 2 } is a connected edge monophonic set and so by Theorem 2.9, m ce (G) = k.Also, it is clear that S 1 is not a connected edge geodetic set of G. Now, we observe that at least one of v i and w i (1 ≤ i ≤ l − k) must belong to every connected edge geodetic set of G. Let S 2 = S 1 ∪ {v 1 , v 2 , . . ., v l−k }.Then for any vertex y / ∈ S 2 , S 2 ∪ {y} is not a connected edge geodetic set of G. Since T = S 2 ∪ {u 1 , u 2 } is a connected edge geodetic set of G, by Theorem 2.10, g ce (G) = l.
If k = 3, then T = {u 1 , u 3 } is an edge monophonic set of G and so m e (G) = 2 = j.Also, no 2-element subset of vertices is a connected edge monophonic set of G.It is clear that T 1 = {u 1 , u 2 , u 3 } is a connected edge monophonic set of G so that m ce (G) = 3 = k.Now, we observe that at least one of v i and w i (1 ≤ i ≤ l − k) must belong to every connected edge geodetic set of G. Let T 2 = {v 1 , v 2 , . . ., v l−3 }.Then for x, y / ∈ T 2 , T 2 ∪ {x, y} is not a connected edge geodetic set of G. Since T 2 ∪ {u 1 , u 2 , u 3 } is a connected edge geodetic set of G, it follows that g ce (G) = l.∈ T , T ∪ {y} is not a connected edge monophonic set of G. Since T ∪ {v 1 , v 3 } is a connected edge monophonic set of G, by Theorem 2.9, m ce (G) = k.It is clear that T is not a connected edge geodetic set of G and it is easily seen that T 1 = T ∪ {v 3 , v 4 , ..., v l−k+4 } is a minimum connected edge geodetic set of G and so by Theorem 2.10, If k = 4, then S 1 = {u 1 , u 2 } is the set of semi-extreme vertices of G. Since S 1 is not an edge monophonic set and since S 2 = S 1 ∪ {v 4 } is an edge monophonic set of G, by Theorem 1.1, we have m e (G) = 3 = j.It is clear that for any vertex y / ∈ S 1 , S 1 ∪{y} is not a connected edge monophonic set of G. Since S 3 = S 1 ∪ {v 1 , v 3 } is a connected edge monophonic set of G, by Theorem 2.9, m ce (G) = 4 = k.Also, it is clear that S 1 is not a connected edge geodetic set of G and it is easily seen that S 4 = S 1 ∪ {v 3 , v 4 , . . ., v l } is a minimum connected edge geodetic set of G and so by Theorem 2.10,

Theorem 2 . 3 .
For any connected graph G of order n, 2 ≤ m e (G) ≤ m ce (G) ≤ n.

Theorem 2 . 7 .Theorem 2 . 8 .
Let G be a connected graph with cut-vertices and S a connected edge monophonic set of G.If v is a cut-vertex of G, then every component of G − v contains an element of S. Every cut-vertex of a connected graph G belongs to every connected edge monophonic set of G.

Corollary 2 . 2 Theorem 2 . 2 Corollary 2 . 1 .
18.There is no non-complete graph G of order n ≥ 3 with m e (G) = 3 and m ce (G) = 3. Corollary 2.19.Let G be any connected graph of order n ≥ 3. Then m e (G) = m ce (G) = 3 if and only if G = K 3 .Theorem 2.20.Let G be a connected graph of order n ≥ 3.If G contains exactly one vertex v of degree n − 1, then every vertex of G other than v is a semi-extreme vertex.Proof.Let v be the unique vertex of degree n − 1 and let w 6 = v be any vertex in G. Then v is adjacent to all the neighbors of w so thatdeg <N(w)> (v) = |N (w)| − 1. Hence w is a semi-extreme vertex of G. 21.Let G be a connected graph of order n ≥ 3.If G contains exactly one vertex v of degree n − 1 and v is not a cut-vertex of G, then m ce (G) = n − 1.In fact, S = V − {v} is the unique minimum connected edge monophonic set of G.Proof.Let v be the unique vertex of degree n−1.Let S = V −{v}.Then by Theorem 2.20, S is the set of semi-extreme vertices of G.By Theorem 2.4, every connected edge monophonic set contains S and so m ce (G) ≥ n−1.Let uv be any edge incident with v. Since v is the only vertex of degree n − 1, there exists at least one vertex w ∈ N (v) such that u and w are nonadjacent.Then uv lies on the monophonic path P : u, v, w with u, w ∈ S. Also, any edge xy not incident with v lies on the x-y monophonic path itself.Thus S is an edge monophonic set of G. Since v is not a cut-vertex of G, the subgraph induced by S is connected so that m ce (G) ≤ n − 1. Hence m ce (G) = n − 1. 22.For the wheelW n = K 1 +C n−1 (n ≥ 5), m ce (W n ) = n−1.The converse of Theorem 2.21 is not true.For the graph G given in Figure 2.3, S = {u 1 , u 2 , u 3 , u 4 , u 5 } is a minimum connected edge monophonic set of G. Therefore, m ce (G) = 5 = n − 1 and no vertex has degree n − 1. Problem 2.23.Characterize graphs G of order n for which m ce (G) = n − Theorem 2.24.Let G be a connected graph of order n ≥ 3.If G has a cut-vertex of degree n − 1, then m ce (G) = n.

2 Theorem 2 . 26 .
and so again deg <N(u)> (u j ) = |N (u)| − 1. Hence u is a semi-extreme vertex of G. Thus every vertex of G is a semi-extreme vertex.For any graph G of order n with at least two vertices of degree n − 1, m ce (G) = n.

Theorem 3 . 1 . 2 Theorem 3 . 2 .
For integers k, l and n with 4 ≤ k ≤ l ≤ n, there exists a connected graph G of order n such that m e (G) = k and m ce (G) = l.Proof.Case 1. 4 ≤ k = l = n.Then, for the compete graph G = K n of order n, by Theorem 1.3 and Corollary 2.6, we have m e (G) = m ce (G) = n.Case 2. 4 ≤ k < l = n.Let G be a tree of order n with k endvertices.Then by Theorem 1.3 and Corollary 2.12, m e (G) = k and m ce (G) = n = l.Case 3. 4 ≤ k < l < n.Let P l−k+3 : u 1 , u 2 , . . ., u l−k+3 be a path of order l − k + 3. Let H be the graph formed by taking n − l + 1 new vertices w 1 , w 2 , . . ., w n−l+1 , and joining each w i (1 ≤ i ≤ n − l + 1) with the vertices u 1 and u 3 in P l−k+3 ; and also joining the vertex w 1 to the vertex u 2 in P l−k+3 .Now, let G be the graph obtained from H by adding k − 4 new vertices y 1 , y 2 , . . ., y k−4 and joining each y i (1 ≤ i ≤ k − 4) with the vertices u 2 and w 1 in H.The graph G has order n and is shown in Figure 3.1.Let S = {w 1 , y 1 , y 2 , . . ., y k−4 , u 2 , u l−k+3 } be the set of semi-extreme vertices of G. Then S is not an edge monophonic set of G. Since S ∪ {u 1 } is an edge monophonic set of G, by Theorem 1.1, m e (G) = k.Now, T = S ∪ {u 3 , u 4 , . . ., u l−k+2 } is the set of semi-extreme vertices and cut-vertices of G.It is clear that T is not a connected edge monophonic set of G. Since T ∪ {u 1 } is a connected edge monophonic set of G, by Theorem 2.9, m ce (G) = l.Case 4. 4 ≤ k = l < n.First let n = k + 1.Let G be a wheel with k + 1 vertices.Then by Theorem 1.3 and Corollary 2.22, we have m e (G) = m ce (G) = k.Next if n > k + 1, then we construct a graph G as follows : Let P 3 : u 1 , u 2 , u 3 be a path of order 3. Let H be the graph formed by taking n − k + 1 new vertices w 1 , w 2 , . . ., w n−k+1 , and joining each w i (1 ≤ i ≤ n − k + 1) with the vertices u 1 and u 3 in P 3 ; and also joining the vertex w 1 to the vertex u 2 in P 3 .Now, let G be the graph obtained from H by adding k − 4 new vertices y 1 , y 2 , . . ., y k−4 and joining each y i (1 ≤ i ≤ k − 4) with the vertices u 2 and w 1 in H.The graph G has order n and is shown in Figure 3.2.Let S = {w 1 , y 1 , y 2 , . . ., y k−4 , u 2 } be the set of semi-extreme vertices of G. Then for any vertex y in G, S ∪ {y} is not an edge monophonic set of G. Since S ∪ {u 1 , u 3 } is an edge monophonic set as well as connected edge monophonic set of G, it follows from Theorems 1.1 and 2.4 that m e (G) = m ce (G) = k.For integers k, l and n with k < l ≤ n and k = 2, 3, there exists a connected graph G of order n such that m e (G) = k and m ce (G) = l.Proof.Case 1. k = 2.If l = n, then let G be a path P n of order n.Hence by Theorems 1.3 and 2.9, m e (G) = 2, m ce (G) = n = l.If l < n, then we construct a graph G as follows: Let P l : u 1 , u 2 , . . ., u l be a path of order l.Let G be the graph obtained from P l by adding n − l new vertices w 1 , w 2 , . . ., w n−l and joining each w i (1 ≤ i ≤ n − l) with u 1 and u 3 in P l .The graph G has order n and is shown in Figure 3.3.If l > 3, then u l is the only semi-extreme vertex of G. Therefore, u l belongs to every edge monophonic set of G. Since S = {u 1 , u l } is an edge monophonic set of G, it follows from Theorem 1.1 that m e (G) = 2 = k.

Case 2 . 2 Theorem 3 . 3 . 2 Theorem 3 . 4 .
k = 3.If l = n, then let G be a tree of order n with three endvertices.Then, by Theorems 1.3 and 2.9, m e (G) = 3 = k and m ce (G) = n = l.If l < n, then we construct a graph G as follows: Let H be a graph obtained from the cycle C 4 : v 1 , v 2 , v 3 , v 4 , v 1 of order 4 and the path P l−3 : u 1 , u 2 , . . ., u l−3 of order l − 3 ≥ 1 by joining u 1 in P l−3 with each v 1 , v 2 and v 3 in C 4 .Let G be the graph obtained from H by adding n−l −1 new vertices w 1 , w 2 , . . ., w n−l−1 and joining each w i (1 ≤ i ≤ n − l − 1) with u 1 and v 4 in H.The graph G has order n and is shown in Figure 3.4.If l > 4, then S 1 = {v 2 , u l−3 } is the set of semi-extreme vertices of G.It is clear that S 1 ∪ {v 4 } is an edge monophonic set so that by Theorem 1.1, m e (G) = 3 = k.Let S 2 = S 1 ∪ {u 1 , u 2 , . . ., u l−4 } be the set of semi-extreme vertices and cut-vertices of G. Then for any vertex y / ∈ S 2 , S 2 ∪ {y} is not a connected edge monophonic set of G.It is easily seen that S 2 ∪ {v 1 , v 4 } is a connected edge monophonic set of G and so by Theorem 2.9, m ce (G) = l.If l = 4, then S 3 = {v 2 , u 1 } is the set of semi-extreme vertices of G. Since S 3 ∪ {v 4 } is an edge monophonic set of G, by Theorem 1.1, m e (G) = 3 = k.It is also easily verified that no 3-element subset of vertices is a connected edge monophonic set of G. Since S 3 ∪ {v 1 , v 4 } is a connected edge monophonic set of G, by Theorem 2.4, m ce (G) = 4 = l.If j, k and l are integers such that 4 ≤ j ≤ k ≤ l, then there exists a connected graph G with m e (G) = j, m ce (G) = k and g ce (G) = l.Proof.Case 1. 4 ≤ j = k = l.Let G = K j be the complete graph of order j.Then by Theorems 1.3, 2.6 and 2.10, m e (G) = m ce (G) = g ce (G) = j.Case 2. 4 ≤ j < k = l.Let G be a tree of order k with j endvertices.Then by Theorem 1.3, m e (G) = j and by Theorems 2.12 and 2.10, m ce (G) = g ce (G) = k.y i (2 ≤ i ≤ k − 3) with both u 2 and u 4 , and also joining u 2 and u 4 .Now, let G be the graph obtained from H by taking (l − k) copies of K 2 with vertex set F i = {v i , w i }(1 ≤ i ≤ l − k) and joining u 1 with each v i (1 ≤ i ≤ l − k) in H and u 3 with each w i (1 ≤ i ≤ l − k) in H.The graph G is shown in Figure 3.6.Let S = {u 2 , u 4 , y 2 , y 3 , . . ., y k−3 } be the set of semi-extreme vertices of G.It is clear that S is not an edge monophonic set of G. Also, for any y / ∈ S, S ∪ {y} is not an edge monophonic set of G. Since S 0 = S ∪ {u 1 , u 3 } is an edge monophonic set as well as a connected edge monophonic set, it follows from Theorems 1.1 and 2.4 that m e (G) = m ce (G) = k.Also, S is not a connected edge geodetic set of G. Now, we observe that at least one vertex of v i and w i (1 ≤ i ≤ l − k) must belong to every connected edge geodetic set of G. Let T = S ∪ {v 1 , v 2 , . . ., v l−k }.Then for any y / ∈ T , T ∪ {y} is not a connected edge geodetic set of G.It follows that T ∪ {u 1 , u 3 } is a minimum connected edge geodetic set of G so that, by Theorem 2.10, g ce (G) = l.For integers j, k and l with j < k ≤ l and j = 2, 3, there exists a connected graph G such that m e (G) = j, m ce (G) = k and g ce (G) = l.Proof.Case 1. j = 2.If k = l, then let G be a path P l of order l.Then by Theorems 1.3, 2.9 and 2.10, m e (G) = 2, m ce (G) = l and g ce (G) = l.If k < l, then we construct a graph G as follows: Let P k : u 1 , u 2 , . . ., u k be a path of order k.Let G be the graph obtained by taking (l − k) copies of K 2 with vertex set F i = {v i , w i }(1 ≤ i ≤ l − k) and joining u 1 with each v i (1 ≤ i ≤ l − k); and also joining u 3 with each w i (1 ≤ i ≤ l − k).The graph G is shown in Figure 3.7.If k > 3, then u k is the only semi-extreme vertex of G. Therefore, u k belongs to every edge monophonic set of G. Since S = {u 1 , u k } is an edge monophonic set of G, it follows from Theorem 1.1 that m e (G) = 2 = j.Let S 1 = {u 3 , u 4 , . . ., u k−1 , u k } be the set of semi-extreme vertices and cutvertices of G.

Case 2 .
j = 3.If k = l, then let G be a tree of order l with three endvertices.Then, by Theorems 1.3, 2.12 and 2.10, m e (G) = 3, m ce (G) = l and g ce (G) = l.If k < l, then we construct a graph G as follows: Let H be a graph obtained from the cycleC 2l−2k+4 : v 1 , v 2 , . . ., v 2l−2k+4 , v 1 of order 2l − 2k + 4,and the path P k−3 : u 1 , u 2 , . . ., u k−3 of order k − 3 by joining u 1 in P k−3 with v 1 , v 2 , v 3 in C 2l−2k+4 .The graph G is shown in Figure 3.8.If k > 4, then S = {v 2 , u k−3 } is the set of semi-extreme vertices of G. Since S is not an edge monophonic set of G and since S ∪ {v 4 } is an edge monophonic set of G, by Theorem 1.1, we have m e (G) = 3 = j.Let T = S ∪ {u 1 , ..., u k−4 } be the set of semi-extreme vertices and cut-vertices of G.It is clear that for any vertex y /