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, diam_{m}{G} of G is diam_{m}{G} = max{e_{m}(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.

**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*.

**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 w_{1} lies on either an *x-z* monophonic path or a *y-z* monophonic path in *G*. Suppose that w_{1} 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 w_{2} 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 Q_{n}
*(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 = {u_{1},u_{2},…,u_{m}} and W = {w_{1}, w_{2},…,w_{n}} 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 = {u_{2},w_{2}}. Then every vertex of *U* lies on a w_{1}-w_{2} monophonic path and every vertex of *W* lies on a _{
u1-u2
} monophonic path. Hence *S* is an *e*-monophonic set of K_{m,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 K_{p} or K_{1,p-1} if and only if _{
mxy(G) = p-2
} for every edge *xy* in *G*.

**Proof.** If G = K_{p}, 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 v_{1},v_{2},…,v_{a-1},w_{1},w_{2},…,w_{b-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.*

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: v_{0},v_{1},v_{2},…,v_{dm} be a monophonic path of length d_{m}. Now, let *S = V(G)-*{v_{1},v_{2},…,v_{dm-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 m_{e}(T) *≤* |S_{1}| = p-d_{m}. If *e = xy* is an edge lies out side *P*, then _{
S2 = S-{x,y}
} is an *e*-monophonic set of *T* so that m_{e}(T) *≤* |S_{2}| = p-d_{m}. 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-d_{m} + 1 = 4. Hence _{
me (G) > p - dm + 1.
}

**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*.

**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
} .

**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
} .

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
} .