ON THE MONOPHONIC NUMBER OF A GRAPH

Abstract

For a connected graph G = (V,E) of order at least two, a set S of vertices of G is a monophonic set of G 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). Certain general properties satisfied by the monophonic sets are studied. Graphs G of order p with m(G) = 2 or p or p - 1 are characterized. For every pair a, b of positive integers with $2{\leq}a{\leq}b$, there is a connected graph G with m(G) = a and g(G) = b, where g(G) is the geodetic number of G. Also we study how the monophonic number of a graph is affected when pendant edges are added to the graph.

1. 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 p and q respec-tively. For basic graph theoretic terminology we refer to Harary [5]. For vertices u and v in a connected graph G, the distance d(u, v) is the length of a shortest u − v path in G. An u − v path of length d(u, v) is called an u − v geodesic. It is known that d is a metric on the vertex set V of G. The neighborhood of a vertex v is the set N(v) consisting of all vertices u which are adjacent with v. The closed neighborhood of a vertex v is the set N[v] = N(v)⋃{v}. A vertex v is an extreme vertex if the subgraph induced by its neighbors is complete. The closed interval I[x, y] consists of all vertices lying on some x − y geodesic of G, while for 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 [1,6] and further studied in [2,4]. The detour distance D(u, v) between two vertices u and v in G is the length of a longest u − v path in G. An u − v path of length D(u, v) is called an u − v detour. It is known that D is a metric on the vertex set V of G. The concept of detour distance was introduced and studied in [3].

FIGURE 1.A graph G with radmG = 3 and diammG = 5

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. For any two vertices u and v in a connected graph G, the monophonic distance dm(u, v) from u to v is defined as the length of a longest u−v monophonic path in G. The monophonic eccentricity em(v) of a vertex v in G is em (v) = max {dm(v, u) : u ∈ V (G)}. The monophonic radius, radmG of G is radmG = min {em(v) : v ∈ V (G)} and the monophonic diameter, diammG of G is diammG = max {em(v) : v ∈ V (G)}. A vertex u in G is monophonic eccentric vertex of a vertex v in G if em(u) = dm(u, v). For the graph G given in Figure 1, d (v1, v4) = 2, D(v1, v4) = 6 and dm (v1, v4) = 4. Thus the monophonic distance is different from both the distance and the detour distance. The usual distance d and the detour distance D are metrics on the vertex set V of a connected graph G, whereas the monophonic distance dm is not a metric on V. For the graph G given in Figure 1, dm(v4, v6) = 5, dm (v4, v5) = 1 and dm(v5, v6) = 1. Hence dm(v4, v6) > dm (v4, v5) + dm(v5, v6) and so the triangle inequality is not satisfied. It is clear that for vertices u and v in a connected graph G of order p, 0≤ d(u, v)≤ dm(u, v)≤ D(u, v)≤ p − 1. The monophonic distance was introduced and studied in [7]. For the graph G given in Figure 1, the monophonic distance between vertices and the monophonic eccentricities of vertices are given in Table 1. Thus radmG = 3 and diammG = 5.

TABLE 1.Monophonic eccentricities of the graph G given in Figure 1

The following theorems will be used in the sequel.

Theorem 1.1 ([6]). Each extreme vertex of a connected graph G belongs to every geodetic set of G.

Theorem 1.2 ([6]). For any tree T with k endvertices, g(T) = k.

Throughout this paper G denotes a connected graph with at least two vertices.

 

2. Monophonic number of a graph

Definition 2.1. A set S of vertices of a graph G is a monophonic set of G if each vertex v of G lies on an x − y monophonic path in G for some x, y ∈ S. The minimum cardinality of a monophonic set of G is the monophonic number of G and is denoted by m(G).

Example 2.2. For the graph G given in Figure 2, S1 = {x,w} and S2 = {u,w} are the minimum monophonic sets of G and so m(G) = 2.

FIGURE 2.A graph G with m(G) = 2

A vertex v in a graph G is a monophonic vertex if v belongs to every minimum monophonic set of G. If G has a unique minimum monophonic set S, then every vertex in S is a monophonic vertex. In the next theorem, we show that there are certain vertices in a nontrivial connected graph G that are monophonic vertices of G.

Theorem 2.3. Each extreme vertex of a connected graph G belongs to every monophonic set of G. Moreover, if the set S of all extreme vertices of G is a monophonic set, then S is the unique minimum monophonic set of G.

Proof. Let u be an extreme vertex and let S be a monophonic set of G. Suppose that u ∉ S. Then u is an internal vertex of an x−y monophonic path, say P, for some x, y ∈ S. Let v and w be the neighbors of u on P. Then v and w are not adjacent and so u is not an extreme vertex, which is a contradiction. Therefore u belongs to every monophonic set of G. The second part of the theorem is clear.

Corollary 2.4. For the complete graph Kp(p ≥ 2), m(Kp) = p.

Theorem 2.5. Let G be a connected graph with a cutvertex v and let S be a monophonic set of G. Then every component of G − v contains an element of S.

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. Since S is a monophonic set, there exists a pair of vertices x and y in S such that u lies in some x − y monophonic path P : x = u0, u1, u2, ..., u, ..., un = y in G with u ≠ x, y. Since v is a cutvertex of G, the x − u subpath P1 of P and the u − y subpath P2 of P both contain v, it follows that P is not a path, which is a contradiction.

Theorem 2.6. No cutvertex of a connected graph G belongs to any minimum monophonic set of G.

Proof. Let v be a cutvertex of G and let S be a minimum monophonic set of G. Then by Theorem 2.5, every component of G − v contains an element of S. Let U and W be two distinct components of G − v and let u ∈ U and w ∈ W. Then v is an internal vertex of an u − w monophonic path. Let S′ = S − {v}. It is clear that every vertex that lies on an u − v monophonic path also lies on an u − w monophonic path. Hence it follows that S′ is a monophonic set of G, which is a contradiction to S a minimum monophonic set of G.

Corollary 2.7. If T is a tree with k endvertices, then m(T) = k.

Proof. This follows from Theorem 2.3 and Theorem 2.6.

We denote the vertex connectivity of a connected graph G by k(G) or k.

Theorem 2.8. If G is a non-complete connected graph such that it has a minimum cutset consisting of k vertices, then m(G) ≤ p − k.

Proof. Since G is a non-complete connected graph, it is clear that 1 ≤ k ≤ p−2. Let U = {u1, u2, u3, ..., uk} be a minimum cutset of G. Let G1, G2, ..., Gr, (r ≥ 2) be the components of G−U and let S = V −U. Then every vertex ui (1 ≤ i ≤ k) is adjacent to at least one vertex of Gj, for each j (1 ≤ j ≤ r). It is clear that S is a monophonic set of G and so m(G) ≤ |S| = p − k.

Remark 2.1. The bound in Theorem 2.8 is sharp. For the cycle C4, m(C4) = 2. Also k = 2 and p − k = 2. Thus m(G) = p − k.

The following theorem is clear.

Theorem 2.9. For any connected graph G, 2 ≤ m(G) ≤ p.

The bounds in the above theorem are sharp. For the complete graph Kp(p ≥ 2), m(Kp) = p. The set of two endvertices of a path Pn(n ≥ 2) is its unique minimum monophonic set so that m(Pn) = 2.

Theorem 2.10. For any integer k such that 2 ≤ k ≤ p there is a connected graph G of order p such that m(G) = k.

Proof. For k = p, the theorem follows from Corollary 2.4 by taking G = Kp. For 2 ≤ k ≤ p−1, the tree G given in Figure 3 has p vertices and it follows from Corollary 2.7 that m(G) = k.

FIGURE 3.The graph G in Theorem 2.10 with m(G) = k

Now we proceed to characterize graphs G for which the bounds in Theorem 2.9 are attained.

Theorem 2.11. For any connected graph G of order p, m(G) = p if and only if G is complete.

Proof. Let m(G) = p. Suppose that G is not a complete graph. Then there exist two vertices u and v such that u and v are not adjacent in G. Since G is connected, there is a monophonic path from u to v, say P, with length at least 2. Let x be a vertex of P such that x ≠ u, v. Then S = V −{x} is a monophonic set of G and hence m(G) ≤ p−1, which is a contradiction. The converse follows from Corollary 2.4.

Definition 2.12. Let x be any vertex in G. A vertex y in G is said to be an x - monophonic superior vertex if for any vertex z with dm(x, y) < dm(x, z), z lies on an x − y monophonic path.

Example 2.13. For any vertex x in the cycle Cp(p ≥ 4), V (Cp) − N[x] is the set of all x - monophonic superior vertices.

We give below a property related with monophonic eccentric vertex of x and x - monophonic superior vertex in a graph G.

Theorem 2.14. Let x be any vertex in G. Then every monophonic eccentric vertex of x is an x - monophonic superior vertex.

Proof. Let y be a monophonic eccentric vertex of x so that em(x) = dm(x, y). If y is not an x - monophonic superior vertex, then there exists a vertex z in G such that dm(x, y) < dm(x, z) and z does not lie on any x−y monophonic path and hence em(x) ≥ dm(x, z) > dm(x, y), which is a contradiction.

Note 2.15. The converse of Theorem 2.14 is not true. For the cycle C6 : v1, v2, v3, v4, v5, v6, v1, the vertex v4 is a v1 - monophonic superior vertex and it is not a monophonic eccentric vertex of v1.

Theorem 2.16. Let G be a connected graph. Then m(G) = 2 if and only if there exist two vertices x and y such that y is an x- monophonic superior vertex and every vertex of G is on an x − y monophonic path.

Proof. Let m(G) = 2 and let S = {x, y} be a minimum monophonic set of G. If y is not an x - monophonic superior vertex, then there is a vertex z in G with dm(x, y) < dm(x, z) and z does not lie on any x−y monophonic path. Thus S is not a monophonic set of G, which is a contradiction. The converse is clear from the definition.

Theorem 2.17. Let G be a connected graph of order p ≥ 3. Then m(G) = p−1 if and only if G = K1 + ⋃ mjKj, where Σ mj ≥ 2.

Proof. Let G = K1 + ⋃ mjKj, where Σ mj ≥ 2. Then G has exactly one cutvertex and all other vertices are extreme and hence by Theorems 2.3 and 2.6, m(G) = p − 1. Conversely, let m(G) = p − 1. Let S be a monophonic set such that |S| = p − 1. Let v ∉ S. We show that v is a cutvertex of G. Otherwise, G − v has just one component. By Theorem 2.3, v is not an extreme vertex of G. Hence there exist vertices x, y ∈ N(v) such that x and y are not adjacent in G − v. Let P be an x − y monophonic path in G − v of length at least 2. Choose a vertex z on P such that z ≠ x, y. Note that z ≠ v. Then it is clear that S1 = V − {v, z} is a monophonic set of G so that m(G) ≤ p − 2, which is a contradiction. Hence v is a cutvertex of G and by Theorem 2.6, v is the only cutvertex of G.

Now, let G1,G2, ...,Gr be the components of G−v. First, we show that each Gi is complete. Suppose that some component, say G1, is not complete. Then there exist two vertices x and y in G1 such that x and y are not adjacent. Choose a vertex z in an x − y geodesic such that z ≠ x, y. Then S2 = V − {v, z} is a monophonic set of G so that m(G) ≤ p − 2, which is a contradiction. Now, it remains to show that v is adjacent to every vertex of Gi for each i (1 ≤ i ≤ r). Otherwise, there exists a component, say Gi, such that v is not adjacent to at least one vertex of Gi. Hence there is a vertex u in Gi such that u is not extreme in G. Then S3 = V −{v, u} is a monophonic set of G so that m(G) ≤ p−2, which is a contradiction. Hence G = K1+⋃mjKj, where K1 = {v} and Σmj ≥ 2.

 

3. Bounds for the monophonic number of a graph

In the following theorem we give an improved upper bound for the mono-phonic number of a graph in terms of its order and monophonic diameter. For convenience, we denote the monoponic diameter diammG by dm itself.

Theorem 3.1. If G is a non-trivial connected graph of order p and monophonic diameter dm, then m(G) ≤ p − dm + 1.

Proof. Let u and v be vertices of G such that dm(u, v) = dm and let P : u = v0, v1, ..., vdm = v be a u − v monophonic path of length dm. Let S = V − {v1, v2, ..., vdm−1}. Then it is clear that S is a monophonic set of G so that m(G) ≤ |S| = p − dm + 1.

For the complete graph Kp(p ≥ 2), dm = 1 and m(Kp) = p so that the bound in Theorem 3.1 is sharp.

A caterpillar is a tree for which the removal of all the endvertices gives a path.

Theorem 3.2. For every non-trivial tree T of order p and monophonic diameter dm, m(T) = p − dm + 1 if and only if T is a caterpillar.

Proof. Let T be any non-trivial tree. Let P : u = v0, v1, ..., vdm be a monophonic diametral path. Let k be the number of endvertices of T and l be the number of internal vertices of T other than v1, v2, ..., vdm−1. Then dm − 1 + l + k = p. By Corollary 2.7, m (T) = k and so m(T) = p−dm−l+1. Hence m(T) = p−dm+1 if and only if l = 0, if and only if all the internal vertices of T lie on the monophonic diametral path P, if and only if T is a caterpillar.

For any connected graph G, radmG ≤ diammG. It is shown in [7] that every two positive integers a and b with a ≤ b are realizable as the monophonic radius and monophonic diameter, respectively, of some connected graph. This theorem can also be extended so that the monophonic number can be prescribed when radmG < diammG.

Theorem 3.3. For positive integers r, d and k ≥ 4 with r < d, there exists a connected graphs G such that radmG = r, diammG = d and m(G) = k.

Proof. We prove this theorem by considering two cases.

Case 1. r = 1. Then d ≥ 2. Let Cd+2 : v1, v2, ..., vd+2, v1 be a cycle of order d+2. Let G be the graph obtained by adding k−2 new vertices u1, u2, ..., uk−2 to Cd+2 and joining each of the vertices u1, u2, ..., uk−2, v3, v4, ..., vd+1 to the vertex v1. The graph G is shown in Figure 4. It is easily verified that 1 ≤ em(x) ≤ d for any vertex x in G and em(v1) = 1, em(v2) = d. Then radmG = 1 and diammG = d. Let S = {u1, u2, ..., uk−2, v2, vd+2} be the set of all extreme vertices of G. Since S is a monophonic set of G, it follows from Theorem 2.3 that m(G) = k.

Case 2. r ≥ 2. Then C : v1, v2, ..., vr+2, v1 be a cycle of order r+2 and let W = K1 + Cd+2 be the wheel with V(Cd+2) = {u1, u2, ..., ud+2}, K1 = {v1} and all other vertices distinct. Now, add k −3 new vertices w1, w2, ..., wk-3 and join each wi(1 ≤ i ≤ k − 3) to the vertex v1 and obtain the graph G of Figure 5. It is easily verified that r ≤ em(x) ≤ d for any vertex x in G and em(v1) = r and em(u1) = d. Thus radmG = r and diammG = d. Let S = {w1, w2, ..., wk-3} be the set of all extreme vertices of G. By Theorem 2.3, every monophonic set of G contains S. It is clear that S is not a monophonic set of G. Let T = S ⋃ {u1, u3, v3}. It is easily verified that T is a minimum monophonic set of G and so m(G) = k.

Figure 4.The graph G in Case 1 of Theorem 3.3

Figure 5.The graph G in Case 2 of Theorem 3.3

Problem 3.4. For any three positive integers r, d and k ≥ 4 with r = d, does there exist a connected graph G with radm = r, diamm = d and m(G) = k?

Theorem 3.5. For each triple d, k, p of integers with 2 ≤ k ≤ p−d+1 and d ≥ 2, there is a connected graph G of order p such that diammG = d and m(G) = k.

Proof. Let Pd+1 : u1, u2, ..., ud+1 be a path of length d. Add p−d−1 new vertices, v1, v2, ..., vk-2, w1, w2, ..., wp-d-k+1 to Pd+1 and join each wi(1 ≤ i ≤ p−d−k+1) to u1, u2 and u3 and also join each vj(1 ≤ j ≤ k-2) to u2, thereby producing the graph G of Figure 6. Then G has order p and monophonic diameter d. If p − d − k + 1 ≤ 1, then S = {v1, v2, ..., vk-2, u1, ud+1} is the set of all extreme vertices of G. Since S is a monophonic set of G, it follows from Theorem 2.3 that m(G) = k. So, let p − d − k + 1 ≥ 2. If d = 2, then S1 = {v1, v2, ..., vk-2} is the set of all extreme vertices of G. It is clear that neither S1 nor S1 ⋃ {x} where x ∉ S1, is a monophonic set of G. Since S2 = S1 ⋃ {u1, u3} is a monophonic set of G, it follows from Theorem 2.3 that m(G) = k. If d ≥ 3, then S3 = {v1, v2, ..., vk-2, ud+1} is the set of all extreme vertices of G. Now, S3 is not a monophonic set of G. Since S4 = S3 ⋃ {u1} is a monophonic set of G, it follows from Theorem 2.3 that m(G) = k.

Figure 6.The graph G in Theorem 3.5 with diammG = d and m(G) = k

Theorem 3.6. For any connected graph G of order p, 2 ≤ m(G) ≤ g(G) ≤ p.

Proof. Since every geodesic is a monophonic path, it follows that every geodetic set is a monophonic set, and hence m(G) ≤ g(G). The other inequalities are trivial.

Remark 3.1. The bounds in Theorem 3.6 are sharp. For the complete graph Kp, m(Kp) = g(Kp) = p. For a non-trivial path Pn, m(Pn) = g(Pn) = 2. Also, if G is a non-trivial tree, or an even cycle, or a complete bipartite graph, then m(G) = g(G). All the inequalities in Theorem 3.6 are strict. For the graph G given in Figure 7, S = {v6, v7, v3} is a minimum monophonic set of G so that m(G) = 3 and no 3-elements subset of the vertex set is a geodetic set of G. Since S ∪ {v1} is a geodetic set of G, it follows that g(G) = 4. Thus we have 2 < m(G) < g(G) < p.

Figure 7.A graph G in Remark 3.1 with 2 < m(G) < g(G) < p

In view of this remark, we have the following problem.

Problem 3.7. Characterize graphs G for which m(G) = g(G)

Theorem 3.8. For every pair a, b of positive integers with 2 ≤ a ≤ b, there is a connected graph G with m(G) = a and g(G) = b.

Proof. For 2 ≤ a = b, any tree with a endvertices has the desired properties, by Theorem 1.2 and Corollary 2.7. So, assume that 2 ≤ a < b. Let Pi : xi,wi, yi (1 ≤ i ≤ b−a) be b−a copies of a path of length 2 and P : v1, v2, v3, v4 a path of length 3. Let G be the graph obtained by joining each xi(1 ≤ i ≤ b−a) in Pi and v2 in P, joining each yi(1 ≤ i ≤ b − a) in Pi and v4 in P; and adding a − 1 new vertices u1, u2, ..., ua−1 and joining each ui(1 ≤ i ≤ a−1) to v4. The graph G is shown in Figure 8. Let S = {vi, ui, ..., ua−1} be the set of all extreme vertices of G. It is easily verified that S is a monophonic set of G and so by Theorem 2.3, m(G) = |S| = a.

Figure 8.The graph G in Theorem 3.8 with m(G) = a and g(G) = b

Next, we show that g(G) = b. By Theorem 1.1, every geodetic set of G contains S. Clearly, S is not a geodetic set of G. It is easily verified that at least one of the vertex of each Pi(1 ≤ i ≤ b − a) must belong to every geodetic set of G. Since T = S ∪ {w1,w2, ...,wb−a} is a geodetic set of G, it follows from Theorem 1.1 that T is a minimum geodetic set of G and so g(G) = b.

 

4. Monophonic number of a graph by adding some pendant edges

Theorem 4.1. If G′ is a graph obtained by adding l pendant edges to a connected graph G, then m(G) ≤ m(G′) ≤ m(G) + l.

Proof. Let G′ be the connected graph obtained from G by adding l pendant edges uivi(1 ≤ i ≤ l), where each ui(1 ≤ i ≤ l) is a vertex of G and each vi(1 ≤ i ≤ l) is not a vertex of G. Let S be a minimum monophonic set of G. Then S ∪ {v1, v2, ..., vl} is a monophonic set of set of G′ and so m(G′) ≤ m(G) + l.

Now, we claim that m(G) ≤ m(G′). Suppose that m(G) > m(G′). Then let S′ be a monophonic set of G′ with |S′| < m (G). Since each vi(1 ≤ i ≤ l) is an extreme vertex of G′, it follows from Theorem 2.3 that {v1, v2, ..., vl} ⊆ S′. Let S = (S′− {v1, v2, ..., vl}) ∪ {u1, u2, ..., ul}. Then S is a subset of V (G) and |S| = |S′| < m(G). Now, we show that S is a monophonic set of G. Let w ∈ V (G)−S. Since S′ is a monophonic set of G′, w lies on an x−y monophonic path P in G′ for some vertices x, y ∈ S′. If neither x nor y is vi(1 ≤ i ≤ l), then x, y ∈ S. If exactly one of x, y is vi(1 ≤ i ≤ l), say x = vi. Then w lies on the ui − y monophonic path in G obtained from P by removing vi. If both x, y ∈ {v1, v2, ..., vl}, then let x = vi and y = vj where i ≠ j. Hence w lies on the ui − uj monophonic path in G obtained from P by removing vi and vj. Thus S is a monophonic set of G. Hence m(G) ≤ |S| < m(G), which is a contradiction.

Remark 4.1. The bounds for m(G′) in Theorem 4.1 are sharp. Consider a tree T with number of endvertices k ≥ 3. Let S = {v1, v2, ..., vk} be the set of all endvertices of T. Then by Corollary 2.7, m(G) = k. If we add a pendant edge to an endvertex of T, then we obtain another tree T′ with k endvertices. Hence m(T) = m(T′). On the otherhand, if we add l pendant edges to a cutvertex of T, then we obtain another tree T′′ with k + l endvertices. Then by Corollary 2.7, m(T′) = m(T) + l.

Now, we proceed to characterize graphs G for which m(G) = m(G′), where G′ is obtained from G by adding l pendant edges.

Theorem 4.2. Let G′ be a graph obtained from a connected graph G by adding l pendant edges uivi(1 ≤ i ≤ l), where ui ∈ V (G) and vi ∉ V (G). Then m(G) = m(G′) if and only if l ≤ m(G) and {u1, u2, ..., ul} is a subset of some minimum monophonic set of G.

Proof. Let l ≤ m(G) and let {u1, u2, ..., ul} be a subset of some minimum monophonic set S of G. Let S′ = (S − {u1, u2, ..., ul}) ⋃ {v1, v2, ..., vl}. Then |S′| = |S|. We show that S′ is a monophonic set of G′. Let z ∈ V (G′) − S′. If z = ui (1 ≤ i ≤ l), then z lies on every vi − w monophonic path in G′, where w ∈ S′, since ui is the only vertex adjacent to vi. So we may assume that z ≠ ui(1 ≤ i ≤ l). Since z is a vertex of G and S is a monophonic set of G, it follows that z lies on some x − y monophonic path P in G for some x, y ∈ S. Then by an argument similar to the one used in the proof of Theorem 4.1, we can show that S′ is a monophonic set of G′. Hence m(G′) ≤ |S′| = |S| = m(G). Now, the result follows from Theorem 4.1.

Conversely, let m(G) = m(G′). Suppose that l > m(G). Since each vi(1 ≤ i ≤ l) is an endvertex of G′, by Theorem 2.3, m(G′) ≥ l. Hence m (G′) > m(G), which is a contradiction. Thus l ≤ m (G′). Now, let S′ be a minimum monophonic set of G′. Since each ui (1 ≤ i ≤ l) is a cutvertex of G′, it follows from Theorem 2.6 that ui ∉ S′ for 1 ≤ i ≤ l. Since each vi(1 ≤ i ≤ l) is an endvertex of G′, it follows from Theorem 2.3 that vi ∈ S′ for 1 ≤ i ≤ l. Let S = (S′ − {v1, v2, ..., vl}) ⋃ {u1, u2, ..., ul}. Then S is a subset of V (G) and |S| = |S′|. Then, as in the proof of Theorem 4.1, S is a monophonic set of G. Since |S| = |S′| = m(G′) = m(G), it follows that S is a minimum monophonic set of G that contains {u1, u2, ..., ul}.

Theorem 4.3. For each triple a, b and l of integers with 2 ≤ a ≤ b, 1 ≤ l ≤ b, and a+l−b ≥ 0, there exists a connected graph G with m(G) = a and m(G′) = b, where G′ is a graph obtained by adding l pendant edges to G.

Proof. Let G be a tree with number of endvertices a. Let G′ be a graph obtained by adding b − a pendant edges to a cutvertex of G and also adding l + a − b pendant edges each with different endvertices of G. Then G′ is another tree with b endvertices. By Corollary 2.7, m(G) = a and m(G′) = b.