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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.36 no.3 Antofagasta Sept. 2017

http://dx.doi.org/10.4067/S0716-09172017000300363 

Articles

Edge fixed monophonic number of a graph

P. Titus1 

S. Eldin Vanaja2 

1 University College of Engineering, Department of Mathematics, Konam, Nagercoil, India, email: titusvino@yahoo.com

2 University College of Engineering, Department of Mathematics, Konam, Nagercoil, India, email: eldinravi28@gmail.com

Abstract:

For an edge xy in a connected graph G of order p ≥ 3, a set S V(G)is an xy-monophonic set of G if each vertex v Є V(G) lies on an x-u monophonic path or a y-u monophonic path for some element u in S. The minimum cardinality of an xy- monophonic set of G is defined as the xy-monophonic number of G, denoted by mxy (G) . An xy-monophonic set of cardinality mxy (G) is called a mxy -set of G. We determine bounds for it and find the same for special classes of graphs. It is shown that for any three positive integers r, d and n ≥ 2 with 2 ≤ r ≤ d, there exists a connected graph G with monophonic radius r, monophonic diameter d and mxy (G) = n for some edge xy in G.

Keywords Monophonic path; vertex monophonic number; edge fixed monophonic number

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 respectively. For basic graph theoretic terminology we refer to (1,2). 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 a simplicial vertex if the subgraph induced by its neighbors is complete. A non-separable graph is connected, non-trivial, and has no cut-vertices. A block of a graph is a maximal non-separable subgraph. A connected block graph} is a connected graph in which each of its blocks is complete. A caterpillar is a tree for which the removal of all the end vertices gives a path.

A chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called it monophonic if it is a chordless path. The closed intervel Im(x,y) consists of all vertices lying on some x-y monophonic of G. 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 (u,v) : u Є V (G)}. The monophonic radius, radm(G) of G is radm {G} = min{em(v):v Є V(G)} and the monophonic diameter, diamm{G} of G is diamm{G} = max{em(v):v Є V(G)}. The monophonic distance was introduced in (3) and further studied in (4). The concept of vertex monophonic number was introduced by Santhakumaran and Titus (5). A set S of vertices of G is an x-monophonic set if each vertex v of G lies on an x-y monophonic path in G for some element y in S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G and is denoted by mx (G) or simply mx . An x-monophonic set of cardinality mx (G) is called a mx - set of G. The following theorems will be used in the sequel.

Theorem 1.1 (2) Let v be a vertex of a connected graph G. The following statements are equivalent:

i) v is a cut-vertex of G.

ii) There exist vertices u and w distinct from v such that v is on every u-w path.

iii) There exists a partition of the set of vertices V - {v} into subsets U and W such that for any vertices u Є U and w Є W, the vertex v is on every u- w path.

Theorem 1.2 (2) Every non-trivial connected graph has at least two vertices which are not cut-vertices.

Theorem 1.3 (2)Let G be a connected graph with at least three vertices. The following statements are equivalent:

i) G is a block.

ii) Every two vertices of G lie on a common cycle.

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

2.Edge fixed monophonic number

Definition 2.1. Let e = xy be any edge of a connected graph G of order at least three. A set S of vertices of G is an xy-monophonic set if every vertex of G lies on either an x-u monophonic path or a y-u monophonic path in G for some element u in S. The minimum cardinality of an xy-monophonic set of G is defined as the xy-monophonic number of G and is denoted by mxy(G) or me(G) . An xy-monophonic set of cardinality mxy(G) is called a mxy -set or me -set of G.

Example 2.2 For the graph G given in Figure 2.1., the minimum edge fixed monophonic sets and the edge fixed monophonic numbers are given in Table 2.1.

Figure 2.1 : G 

Theorem 2.3. For any edge xy in a connected graph G of order at least three, the vertices x and y do not belong to any minimum xy-monophonic set of G.

Proof. Suppose that x belongs to a minimum xy-monophonic set, say S, of G. Since G is a connected graph with at least three vertices and xy in an edge, it follows from the definition of an xy-monophonic set that S contains a vertex v different from x and y. Since the vertex x lies on every x-v monophonic path in G, it follows that T = S-{x} is an xy-monophonic set of G, which is a contradiction to S a minimum xy-monophonic set of G. Similarly, y does not belong to any minimum xy-monophonic set of G.

Table 2.1: The Edge Fixed Monophonic Number of a Graph 

Theorem 2.4. Let xy be any edge of a connected graph G of order at least three. Then

i) every simplicial vertex of G other than the vertices x and y (whether x or y is simplicial or not) belongs to every mxy- set.

ii) no cut-vertex of G belongs to any mxy- set.

Proof. (i) By Theorem 2.3, the vertices x and y do not belong to any mxy- set. So, let u ≠ x,y be a simplicial vertex of G. Let S be a mxy -set of G such that u S. Then u is an internal vertex of either an x-v monophonic path or a y-v monophonic path for some element v in S. Without loss of generality, let P be an x-v monophonic path with u is an internal vertex. Then both the neighbors of u on P are not adjacent and hence u is not a simplicial vertex, which is a contradiction.

(ii) Let v be a cut-vertex of G. Then by Theorem 1.1, there exists a partition of the set of vertices V- {v} into subsets U and W such that for any vertex u Є U and w Є W, the vertex v lies on every u-w path. Let S be a mxy-set of G. We consider three cases.

Case (i): Both x and y belong to U. Suppose that S ∩ W = Ø. Let w1 Є W. Since S is an xy-monophonic set, there exists an element z in S such that w1 lies on either an x-z monophonic path or a y-z monophonic path in G. Suppose that w1 lies on an x-z monophonic path P:x = z0, z1,…, w1,…,zn = z in G . Then the x-w1 subpath of P and w1-z subpath of P both contain v so that P is not a path in G, which is a contradiction. Hence S ∩ W = Ø. Let w2 Є S ∩ W. Then v is an internal vertex of any x-w2 monophonic path and v is also an internal vertex of any y-w2 monophonic path. If v Є S, then let S' = S-{v}. It is clear that every vertex that lies on an x-v monophonic path also lies on an x-w2 monophonic path. Hence it follows that S' is an xy-monophonic set of G, which is a contradiction to S a minimum xy-monophonic set of G. Thus v does not belong to any minimum xy-monophonic set of G.

Case (ii) : Both x and y belong to W. It is simillar to Case(i).

Case (iii) : Either x = v or y = v. By Theorem 2.3, v does not belong to any mxy-set .

Corollary 2.5. Let T be a tree with k end vertices. Then mxy (T) = k - 1 or k according as xy is an end edge or cut-edge.

Proof. This follows from Theorem 2.4.

Corollary 2.6. Let K1,n (n ≥ 2) be a star. Then mxy (K1,n) = n - 1 for any edge xy in K1,n.

Corollary 2.7. Let G be a complete graph Kp (p ≥ 3) . Then mxy (G) = p - 2 for any edge xy in G.

Theorem 2.8.For any edge xy in the cube Qn (n ≥ 3), mxy (Qn) = 1 .

Proof. Let e = xy be an edge in Qn and let x = (a1,a2,…,an) , where ai Є {0,1} . Let x' = (a’1,a’2,…,a’n ) be another vertex of Qn such that a’1 is the compliment of ai . Let u be any vertex in Qn . For convenience, let u = (a1,a’2,a3…,an) . Then u lies on an x-x' monophonic path $P: x = (a1,a2,…,an) , (a1,a’2,a3,…,an) ,…, (a’1,a’2,…,a’n-1,an) , (a’1,a’2,…,a’n ) = x'. Hence {x'} is an xy-monophonic set of Qn and so mxy(Qn) = 1.

Theorem 2.9. i) For any edge xy in the wheel Wn = K1 + Cn-1 (n ≥ 5), mxy (Wn) = 1.

ii) For any edge xy in the complete bipartite graph Km,n (1 ≤ m ≤ n),

Proof. (i) Let xy be an edge in Wn . Then either x or y is a vertex of Cn-1 . Let x Є V (Cn-1) and let z be a non-adjacent vertex of x in Cn-1. It is clear that every vertex of Wn lies on an x - z monophonic path. Hence {z} is a mxy -set of Wn and so mxy (Wn) = 1 .

(ii) Let U = {u1,u2,…,um} and W = {w1, w2,…,wn} be the vertex subsets of the bipartition of the vertices of Km,n . If m = 1, then by Corollary 2.6, mxy(K1,n) = n-1 for any edge xy in K1,n . If m = 2, let e be an edge in Km,n , say e = u1w1 . It is clear that every vertex of Km,n lies on an u1-u2 monophonic path. Hence {u2} is an e-monophonic set of Km,n and so me(Km,n) = 1 . If m ≥ 3, then it is clear that no singleton subset of V is an e-monophonic set of Km,n and so me(Km,n) ≥ 2 . Without loss of generality, take e = u1w1 . Let S = {u2,w2}. Then every vertex of U lies on a w1-w2 monophonic path and every vertex of W lies on a u1-u2 monophonic path. Hence S is an e-monophonic set of Km,n and so me(Km,n) = 2 .

Theorem 2.10. For any edge xy in a connected graph G of order p ≥ 3, 1 ≤ mxy(G) ≤ p -2.

Proof. It is clear from the definition of mxy-set that mxy(G) ≥ 1 . Also, since the vertices x and y do not belong to any mxy-set , it follows that mxy(G) ≤ p-2.

Remark 2.11. The bounds for mxy(G) in Theorem 2.10 are sharp. If C is any cycle, then mxy(C) = 1 for any edge xy in C. For any edge xy in a complete graph Kp (p 3), mxy(Kp) = p-2.

Now we proceed to characterize graphs for which the upper bound in Theorem 2.10 is attained.

Theorem 2.12. Let G be a connected graph of order at least 3. Then G is either Kp or K1,p-1 if and only if mxy(G) = p-2 for every edge xy in G.

Proof. If G = Kp, then by Corollary 2.7, mxy(G) = p-2 for every edge xy in G. If G = K1,p-1 , then by Corollary 2.6, mxy(G) = p-2 for any edge xy in G. Conversely, suppose that mxy(G) = p-2 for every edge xy in G. By Theorem 1.2, G has at least two vertices which are not cut-vertices. Let xy be an edge of G with x is not a cut-vertex. If G has two or more cut-vertices, then by Theorem 2.4(ii), mxy(G) ≤ p-3 , which is a contradiction. Thus the number of cut-vertices k of G is at most one.

Case (i) k = 0. Then the graph G is a block. Now we claim that G is complete. If G is not complete, then there exist two vertices x and y in G such that d(x,y) ≥ 2. By Theorem 1.3, x and y lie on a common cycle and hence x and y lie on a smallest cycle C:x,x1,x2,…,y,…,xn,x of length at least 4. Then (V(G)-V(C))U {y} is an xx1 -monophonic set of G and so mxx1(G) ≤ p-3 , which is a contradiction. Hence G is the complete graph.

Case (ii) k = 1. Let x be the cut-vertex of G. If p = 3, then G = P 3, a star with three vertices. If p ≥ 4, we claim that G = K1,p-1 . It is enough to prove that degree of every vertex other than x is one. Suppose that there exists a vertex, say y, with deg y ≥ 2. Let z ≠ x be an adjacent vertex of y in G. Let e = yz. Since the vertices y and z do not lie on any minimum yz-monophonic set of G and by Theorem 2.4(ii), we have myz(G) ≤ p-3 , which is a contradiction. Thus every vertex of G other than x is of degree one. Hence G is a star.

Theorem 2.13.For any edge xy in a connected graph G, every x -monophonic set of G is an xy -monophonic set of G.

Proof. Let S be an x-monophonic set of G. Then every vertex of G lies on an x-z monophonic path for some z in S. It follows that S is an xy-monophonic set of G.

Corollary 2.14.For any edge xy in a connected graph G, mxy(G) ≤ min{mx(G),my(G)}.

Theorem 2.15.For every pair a,b of integers with 1 ≤ a ≤ b, there is a connected graph G with mxy(G) = a and mx(G) = b and for some edge xy in G.

Proof. Let C4:x,y,z,u,x be a cycle of order 4. Add b-1 new vertices v1,v2,…,va-1,w1,w2,…,wb-a and joining each vi (1 ≤ i ≤ a-1) to x and joining each wj(1 ≤ j ≤ b-a) to the vertices y and u, thereby producing the graph G given in Figure 2.2. Let S = {v1,v2,…, va-1} be the set of all simplicial vertices of G. Since S is not an xy-monophonic set, it follows from Theorem 2.4(i) that mxy(G) ≥ a . On the other hand, S1 = S U {u} is an xy-monophonic set of G and so mxy(G) = | S1 | = a. Clearly, S2 = {v1,v2,…,va-1,z,w1,w2,…,wb-a} is the unique x-monophonic set of G and so mx(G) = | S2 | = b.

Figure 2.2 : G 

We have seen that if G is a connected graph of order p ≥ 3, then 1 ≤ mxy(G) ≤ p-2 for any edge xy in G. In the following theorem we give an improved upper bound for the edge fixed monophonic number of a tree in terms of its order and monophonic diameter.

Theorem 2.16. If T is a tree of order p and monophonic diameter dm , then mxy (T) ≤ p - dm + 1 for any edge xy in T.

Proof. Let P: v0,v1,v2,…,vdm be a monophonic path of length dm. Now, let S = V(G)-{v1,v2,…,vdm-1}. If e is an internal edge of P, then clearly S is an e-monophonic set of T so that me(T) ≤ |S| = p-dm+1 . If e is an end edge of P, say e = v0v1 , then S1 = S-{v0} is an e-monophonic set of T so that me(T) |S1| = p-dm. If e = xy is an edge lies out side P, then S2 = S-{x,y} is an e-monophonic set of T so that me(T) |S2| = p-dm. Hence for any edge xy in T, mxy(T) ≤ p-dm+1.

Remark 2.17. The bound in Theorem 2.16 is not true for any graph. For example, consider the graph G given in Figure 2.3. Here p = 7, dm(G) = 4, me(G) = 5 and p-dm + 1 = 4. Hence me (G) > p - dm + 1.

Figure 2.3 : G 

Theorem 2.18.For any edge xy in a non-trivial tree T of order p and monophonic diameter dm, mxy (T) = p - dm or p-dm + 1 , if and only if T is a caterpillar.

Proof. Let T be any non-trivial tree. Let P:v0,v1,…,vd be a monophonic path of length dm . Let k be the number of end vertices of T and let l be the number of internal vertices of T other than v1,v2,…,vd-1 . Then dm-1+l+k = p . By Corollary 2.5, mxy(T) = k or k-1 for any edge xy in T and so mxy(T) = p-dm-l+1 or p-dm-l for any edge xy in T. Hence mxy(T) = p-dm+1 or p-dm for any edge xy in T if and only if l =0 , if and only if all the internal vertices of T lie on the monophonic path P, if and only if T is a caterpillar.

For any connected graph G, radm(G) ≤ diamm(G) . It is shown in ((3)) 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 result can be extended so that the edge fixed monophonic number can be prescribed.

Theorem 2.19. For positive integers r,d and n ≥ 2 with 2 ≤ r ≤ d, there exists a connected graph G with radm(G) = r , diamm(G) = d and mxy(G) = n for some edge xy in G.

Proof. Case (i) 2 ≤ r = d. Let Cr+2: v1, v2,…, vr+2, v1 be the cycle of order r + 2. Let G be the graph obtained from Cr+2 by adding n vertices u1,u2,…,un and joining each vertex ui (1 ≤ I ≤ n) to both v2 and vr+2 , and also adding the edge v1u1 . The graph G is shown in Figure 2.4. It is easily verified that the monophonic eccentricity of each vertex of G is r and so radm(G) = diamm(G) = r . Also, for the edge v1u1 , it is clear that S = {vr+1,u2,…,un} is a minimum xy-monophonic set of G and so mxy(G) = n.

Figure 2.4 : G 

Case (ii) 2 ≤ r < d ≤ 2r. Let Cr+2 = v1,v2,…, vr+2 ,v1 be the cycle of order r + 2 and let Pd-r+1 : u0,u1,…,ud-r be a path of order d-r+1. Let H be the graph obtained from Cr+2 and Pd-r+1 by identifying v1 in C r+2 and u0 in Pd-r+1 . Let G be the graph obtained from H by adding n-1 new vertices w1,w2,…,wn-1 and joining each wi (1 ≤ i ≤ n-1) with ud-r-1 . The graph G is shown in Figure 2.5. It is easily verified that r ≤ em(x) ≤ d for any vertex x in G, em(v1) = r and em(v3) = d . Thus radm(G) = r and diamm(G) = d . For the edge e = ud-r-1ud-r , S = {w1,w2,…,wn-1,v3} is a minimum e-monophonic set of G and so me(G) = n .

Figure 2.5 : G 

Case (iii) d > 2r. Let P2r-1:v1,v2,...,v2r-1 be a path of order 2r-1. Let G be the graph obtained from the wheel W = K1+Cd+2 and the complete graph Kn by identifying the vertex v1 of P2r-1 with the central vertex of W, and identifying the vertex v2r-1 of P2r-1 with a vertex of Kn . The graph G is shown in Figure 2.6.. Since d > 2r, we have em(x) = d for any vertex x Є V(Cd+2) . Also, em(x) = 2r for any vertex x Є V(Kn)- v2r-1 ; r ≤ em(x) ≤ 2r-1 for any vertex x Є V(P2r-1) ; and em(x) = r for the central vertex x of P2r-1 . Thus radm (G) = r and diamm (G) = d .

Figure 2.6 : G 

Let S = V(Kn)-{v2r-1} be the set of all simplicial vertices of G. Then by Theorem 2.4(i), every me -set contains S for the edge e = u1u2 . It is clear that S is not an e-monophonic set of G and so me(G)> |S| = n - 1. Since S' = S U {ud+1} is an e-monophonic set of G, we have me(G) = n .

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Received: July 2015; Accepted: May 2017

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