1. Introduction

By a graph G = (V, E) we mean a finite undirected connected graph without loops and multiple edges. The order and size of G are denoted by p and q, respectively. For basic graph theoretic terminology we refer ^{(1, 3)}. The distance d(x, y) between two vertices x and y in a connected graph G 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. Let G_{
1
} = (V_{
1
} , E_{
1
} ) and G_{
2
} = (V_{
2
} , E_{
2
} ) be two graphs with V_{
1
} ∩ V_{
2
} = φ, then the join G_{
1
} +G_{
2
} is a graph G = (V, E), where V = V_{
1
} ∪ V_{
2
} and E = E_{
1
} ∪ E_{
2
} together with all the edges joining vertices of V_{
1
} to vertices of V_{
2
} and m_{
j
} K_{
j
} denotes m_{
j
} -copies of the complete graph K_{
j
} .

A chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A longest x − y monophonic path is called an x − y detour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G lies on an x − y detour monophonic path for some x and y in S. The mínimum 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 ^{7} and further studied in ^{6}.

A connected detour monophonic set of G is a detour monophonic set S such that the subgraph G[S] induced by S is connected. The mínimum cardinality of a connected detour monophonic set of G is the connected detour monophonic number of G and is denoted by dm_{
c
} (G). The connected detour monophonic number of a graph was introduced and studied in ^{8} .

The detour monophonic concepts have interesting applications in Channel Assignment Problem in radio technologies. Also, there are useful applications of these concepts to security based communication network design.

For any two vertices u and v in a connected graph G, the monophonic distance d_{
m
} (u, v) from u to v is defined as the length of a longest u - v monophonic path in G. The monophonic eccentricity e_{
m
} (v) of a vertex v in G is e_{
m
} (v) = max {d_{
m
} (v, u) : u ∈ V (G)}. The monophonic radius, rad_{
m
} (G) of G is rad_{
m
} (G) = min {e_{
m
} (v) : v ∈ V (G)} and the monophonic diameter, diam_{
m
} (G) of G is diam_{
m
} (G) = max {e_{
m
} (v) : v ∈ V (G)}. The monophonic distance was introduced in ^{4} and further studied in ^{5} .

The following theorems will be used in the sequel.

**Theorem 1.1**. ^{7} Each extreme vertex of a connected graph G belongs to every detour monophonic set of G. Moreover, if the set S of all extreme vertices of G is a detour monophonic set, then S is the unique mínimum detour monophonic set of G.

**Theorem 1.2**. ^{7} Let G be a connected graph with a cut-vertex v and let S be a detour monophonic set of G. Then every component of G - v contains an element of S.

**Theorem 1.3.**^{8} Each extreme vertex of a connected graph G belongs to every connected detour monophonic set of G.

**Theorem 1.4**. ^{8} Every cut-vertex of a connected graph G belongs to every connected detour monophonic set of G.

**Theorem 1.5**. ^{8} For the complete graph K_{
p
} (p ≥ 2), dm_{
c
} (K_{
p
} ) = p.

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

2. Total detour monophonic number

**Definition 2.1**. A total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph G[S] induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dm_{
t
} (G). A total detour monophonic set of cardinality dm_{
t
} (G) is called a dm_{
t
} -set of G.

Obviously, any connected detour monophonic set of G is a total detour monophonic set of G.

**Example 2.2**. For the graph G in Figure 2.1, it is clear that S = {v_{
1
} , v_{
4
} , v_{
5
} } is a minimum detour monophonic set of G so that dm(G) = 3. It is easily verified that the set S_{
1
} = {v_{
1
} , v_{
2
} , v_{
4
} , v_{
5
} } is a minimum total detour monophonic set of G so that dm_{
t
} (G) = 4. Also, it is clear that S_{
2
} = {v_{
1
} , v_{
2
} , v_{
3
} , v_{
4
} , v_{
5
} } is a minimum connected detour onophonic set of G and so dm_{
c
} (G) = 5. Thus the detour monophonic number, total detour monophonic number and connected detour monophonic number of a graph all different.

It was shown in ^{2} that determining the monophonic number of a graph is an NP-hard problem. Since every total detour monophonic set of G is a monophonic set, determining the total detour monophonic number of a graph is also an NP-hard problem.

**Definition 2.3**. A vertex v of a connected graph G is called a support vertex of G if it is adjacent to an end-vertex of G.

**Theorem 2.4**. Each extreme vertex and each support vertex of a connected graph G belongs to every total detour monophonic set of G. If the set S of all extreme vertices and support vertices form a total detour monophonic set, then it is the unique minimum total detour monophonic set of G.

**Proof**. Since every total detour monophonic set of G is a detour monophonic set of G, by Theorem 1.1, each extreme vertex belongs to every total detour monophonic set of G. Since a total detour monophonic set of G contains no isolated vertices, it follows that each support vertex of G also belongs to every total detour monophonic set of G. Thus, if S is the set of all extreme vertices and support vertices of G, then dm_{
t
} (G) ≥ |S|. On the other hand, if S is a total detour monophonic set of G, then dmt(G) ≤ |S|. Therefore dm_{
t
} (G) = |S| and S is the unique minimum total detour monophonic set of G.

**Corollary 2.5**. For the complete graph K_{
p
} (p ≥ 2), dm_{
t
} (G) = p.

**Theorem 2.6**. Let G be a connected graph with cut-vertices and let S be a total detour monophonic set of G. If v is a cut-vertex of G, then every component of G − v contains an element of S.

**Proof**. Since every total detour monophonic set of G is a detour monophonic set of G, the result follows from Theorem 1.2.

**Theorem 2.7**. For a connected graph G of order p, 2 ≤ dm(G) ≤ dm_{
t
} (G) ≤ dm_{
c
} (G) ≤ p.

**Proof**. Any detour monophonic set of G needs at least two vertices and so dm(G) ≥ 2. Since every total detour monophonic set of G is also a detour monophonic set of G, it follows that dm(G) ≤ dm_{t}(G). Also, since every connected detour monophonic set of G is a total detour monophonic set of G, it follows that dm_{
t
} (G) ≤ dm_{
c
} (G). Since V (G) is a connected detour monophonic set of G, it is clear that dm_{
c
} (G) ≤ p. Hence 2 ≤ dm(G) ≤ dm_{
t
} (G) ≤ dm_{
c
} (G) ≤ p.

**Corollary 2.8**. Let G be a connected graph. If dm_{
t
} (G) = 2, then dm(G) =2.

For any non-trivial path of order at least 4, the detour monophonic number is 2 and the total detour monophonic number is 4. This shows that the converse of the Corollary 2.8 need not be true.

**Remark 2.9**. The bounds in Theorem 2.7 are sharp. For the omplete graph G = K_{
2
} , dm_{
t
} (G) = 2 and for the complete graph K_{
p
} , dm_{
t
} (K_{
p
} ) = p. For the graph G given in Figure 2.1, dm(G) = 3, dm_{
t
} (G) = 4, dm_{
c
} (G) = 5 and p = 8 so that 2 < dm(G) < dm_{
t
} (G) < dm_{
c
} (G) < p. Hence all the parameters in Theorem 2.7 are distinct.

**Theorem 2.10**. For any non-trivial tree T , the set of all end-vertices and support vertices of T is the unique minimum total detour monophonic set of G.

**Proof.** Since the set of all end-vertices and support vertices of T forms a total detour monophonic set, the result follows from Theorem 2.4.

Now we proceed to characterize graphs G for which the lower bound in Theorem 2.7 is attained.

**Theorem 2.11**. For any connected graph G, dmt(G) = 2 if and only if G = K_{
2
} .

**Proof**. If G = K_{
2
} , then dm_{
t
} (G) = 2. Conversely, let dm_{
t
} (G) = 2. Let S = {u, v} be a minimum total detour monophonic set of G. Then uv is an edge. It is clear that a vertex different from u and v cannot lie on a u - v detour monophonic path and so G = K_{
2
} .

**Theorem 2.12**. Let G be a connected graph with at least 2 vertices. Then dm_{
t
} (G) ≤ 2 dm(G).

**Proof**. Let S = {v_{
1
} , v_{
2
} , . . . , v_{
k
} } be a minimum detour monophonic set of G. Let u_{
i
} ∈ N (v_{
i
} ) for i = 1, 2, . . . , k and let T = {u_{
1
} , u_{
2
} , . . . , u_{
k
} }. Then S ∪ T is a total detour monophonic set of G so that dm_{
t
} (G) ≤ |S ∪ T | ≤ 2k = 2dm(G).

**Theorem 2.13**. If G = C_{
p
} or G =
, where H is a graph of order p − 2, then dm_{
t
} (G) = 3.

**Proof**. First, suppose that G = C_{
p
} . It is easily verified that any three consecutive vertices of C_{
p
} is a minimum total detour monophonic set of C_{
p
} and so dm_{
t
} (G) = dmt(C_{
p
} ) = 3. Next, suppose that
, where H is a graph of order p − 2. Let
. Then for any vertex v of H, the set S = {v, u_{
1
} , u_{
2
} } is a minimum total detour monophonic set of G and so dm_{t}(G) = 3.

**Theorem 2.14**. For the complete bipartite graph G = K_{
r,s
} (2 ≤ r ≤ s),

**Proof.** Let U = {u_{
1
} , u_{
2
} , . . . , u_{
r
} } and W = {w_{
1
} , w_{
2
} , . . . , w_{
s
} } be the bipartition of G. We prove this theorem by considering four cases.

**Case 1**. 2 = r = s. Then G is the cycle C_{
4
} and by Theorem 2.13, dm_{
t
} (G) = 3.

**Case 2**. 2 = r < s. Then the minimum total detour monophonic set of G is obtained by choosing the two elements from U and any one element from W and so dm_{
t
} (G) = 3.

**Case 3**. 3 ≤ r ≤ s. If 3 = r = s, then any minimum total detour monophonic set of G is of the following forms:
for some
for some
a set containing any two elements from each of U and W. If 3 = r < s, then any minimum total detour monophonic set of G is either U ∪ {w_{
j
} } for some j(1 ≤ j ≤ s), or any set containing any two elements from each of U and W. Hence in both cases, we have dm_{
t
} (G) = 4.

**Case 4**. 4 ≤ r ≤ s. Then any minimum total detour monophonic set of G contains any two elements from each of U and W, and hence dm_{
t
} (G) = 4.

**Theorem 2**.15. If
, then dm_{
t
} (G) = p − 1.

**Proof**. Let
. It is clear that G has exactly one cut-vertex which is not the support vertex and all the remaining vertices are extreme vertices. Hence by Theorem 2.4, we have dm_{
t
} (G) = p − 1.

**3. Some realization results on the total detour monophonic Number**

For any connected graph G, rad_{
m
} (G) ≤ diam_{
m
} (G). It is shown in ^{4} 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 total detour monophonic number can be prescribed when rad_{
m
} (G) < diam_{
m
} (G).

**Theorem 3.1**. For positive integers r, d and k ≥ 6 with r < d, there exists a connected graph G such that rad_{
m
} (G) = r, diam_{
m
} (G) = d and dm_{
t
} (G) = k.

**Proof**. We prove this theorem by considering two cases.

**Case 1**. r = 1. Then d ≥ 2. Let C_{
d+2
}: v_{
1
} , v_{
2
} , . . . , v_{
d+2
} , v_{
1
} be a cycle of order d + 2. Let G be the graph obtained by adding k − 3 new vertices u_{
1
} , u_{
2
} , . . . , u_{
k−3
} to C_{
d+2
} and joining each of the vertices u_{
1
} , u_{
2
} , . . . , u_{
k−3
} , v_{
3
} , v_{
4
} , . . . , v_{
d+1
} to the vertex v_{
1
} . The graph G is shown in Figure 3.1. It is easily verified that 1 ≤ e_{
m
} (x) ≤ d for any vertex x in G and e_{
m
} (v_{
1
} ) = 1, e_{
m
} (v_{
2
} ) = d. Then rad_{
m
} (G) = 1 and diam_{
m
} (G) = d.

Let S = {u_{
1
} , u_{
2
} , . . . , u_{
k−3
} , v_{
2
} , v_{
d+2
} , v_{
1
} } be the set of all extreme vertices and support vertex of G. By Theorem 2.4, S is a subset of any total detour monophonic set of G. It is clear that S is the total detour monophonic set of G and so dm_{
t
} (G) = k.

**Case 2**. r ≥ 2. Let C : v_{
1
} , v_{
2
} , . . . , v_{
r+2
} , v_{
1
} be a cycle of order r + 2 and let W = K_{1} + C_{
d+2
} be the wheel with V (C_{
d+2
} ) = {u_{
1
} , u_{
2
} , . . . , u_{
d+2
} }. Let H be the graph obtained from C and W by identifying v_{
1
} of C and the central vertex K_{
1
} of W. If d is odd, then add k − 6 new vertices w_{
1
} , w_{
2
} , . . . , w_{
k−6
} to the graph H and join each w_{i}(1 ≤ i ≤ k − 6) to the vertex v_{
1
} and obtain the graph G of Figure 3.2. It is easily verified that r ≤ e_{
m
} (x) ≤ d for any vertex x in G and
. Thus rad_{
m
} (G) = r and diam_{
m
} (G) = d. Let S = {w_{
1
} , w_{
2
} , . . . , w_{
k−6
} , v_{1}} be the set of all extreme vertices and support vertex of G. By Theorem 2.4, every total detour monophonic set of G contains S. It is clear that S is not a total detour monophonic set of G. Also, S ∪ {x_{
1
} , x_{
2
} , x_{
3
} , x_{
4
} } where x_{j}(1 ≤ j ≤ 4) ∈ V (G) − S, is not a total detour monophonic set of G. Let T = S ∪ {u_{
1
} , u_{
2
} , u_{
3
} , v_{
2
} , v_{
3
} }. It is easily verified that T is a total detour monophonic set of G and so dm_{
t
} (G) = k. If d is even, then add k − 5 new vertices w_{
1
} , w_{
2
} , . . . , w_{
k−5
} to the graph H and join each w_{i}(1 ≤ i ≤ k − 5) to the vertex v_{
1
} and obtain the graph G. Similar to the above argument,
is a minimum total detour monophonic set of G and so dm_{t}(G) = k.

**Problem 3.2**. For any three positive integers r, d and k ≥ 6 with r = d does there exist a connected graph G with radm(G) = r, diamm(G) = d and dmt(G) = k?

**Theorem 3.3**. If p, d and k are positive integers such that 2 ≤ d ≤ p − 2, 3 ≤ k ≤ p and p − d − k +3 ≥ 0, then there exists a connected graph G of order p, monophonic diameter d and dm_{t}(G) = k.

**Proof**. We prove this theorem by considering two cases.

**Case 1**. d = 2. First, let k = 3. Let P_{
3
}: v_{
1
} , v_{
2
} , v_{
3
} be the path of order 3. Now, add p − 3 new vertices w_{
1
} , w_{
2
} , ..., w_{
p−3
} to P_{
3
} . Let G be the graph obtained by joining each w_{
i
} (1 ≤ i ≤ p − 3) to v_{
1
} and v_{
3
} . The graph G is shown in Figure 3.3. Then G has order p and monophonic diameter d = 2. Clearly S = {v_{
1
} , v_{
2
} , v_{
3
} } is a minimum total detour monophonic set of G so that dmt(G) = k = 3.

Now, let 4 ≤ k ≤ p. Let K_{
p−1
} be the complete graph with the vertex set
. Now, add the new vertex x to K_{
p−1
} and let G be the graph obtained by joining x with each vertex w_{
i
} (1 ≤ i ≤ p − k + 1). The graph G is shown in Figure 3.4. Then G has order p and monophonic diameter d = 2. Let
be the set of all extreme vertices of G. By Theorem 2.4, every total detour monophonic set of G contains S. It is clear that S is a detour monophonic set of G. Since the induced subgraph
has an isolated vertex, dm_{
t
} (G) ≥ k. For any vertex
, it is clear that S ∪{v} is a minimum total detour monophonic set of G and so dm_{
t
} (G) = k.

**Case 2**. 3 ≤ d ≤ p −2. First, let k = 3. Let
be the cycle of order d+2. Add p − d − 2 new vertices w_{
1
} , w_{
2
} , ..., w_{
p−d−2
} to C and join each vertex w_{
i
} (1 ≤ i ≤ p −d −2) to both v_{
1
} and v_{
3
} , thereby producing the graph G of Figure 3.5. Then G has order p and monophonic diameter d. It is clear that S = {v_{
3
} , v_{
4
} , v_{
5
} } is a minimum total detour monophonic set of G and so dm_{
t
} (G) = 3 = k.

Now, let k ≥ 4. Let
be a path of order d. Add p −d new vertices
to P_{
d
} and join
to both v_{
0
} and v_{
2
} ; and join
, thereby producing the graph G of Figure 3.6..

Then G has order p and monophonic diameter d. Let
be the set of all end-vertices and support vertex of G. By Theorem 2.4, every total detour monophonic set of G contains S. It is clear that S is not a total detour monophonic set of G. Also, for any
is not a total detour monophonic set of G. It is easily seen that
is a minimum total detour monophonic set of G and so dm_{
t
} (G) = k.

In view of Theorems 2.13, 2.14 and 3.3, we leave the following problema as an open question.

**Problem 3.4.** Characterize graphs for G which dm_{
t
} (G).

In view of Theorem 2.7, we have the following realization result.

**Theorem 3.5**. If a,b and p are positive integers such that 4 ≤ a ≤ b ≤ p, then there exists a connected graph G of order p, dm_{
t
} (G) = a and dm_{
c
} (G) = b.

**Proof.** We prove this theorem by considering four cases.

**Case 1**. 4 ≤ a = b = p. Let G = K_{
p
} . Then by Corollary 2.5 and Theorem 1.5, we have dm_{
t
} (G)= dm_{
c
} (G)= p.

**Case 2**. 4 ≤ a < b < p. Let
be a path of order 𝑏−𝑎+3. Add 𝑝−𝑏+𝑎−3 new vertices
with both u_{
1
} and u_{
3
} ; and also join each of 𝑤 𝑖 (1≤𝑖≤𝑎−3) with u_{
b−a+3
} , thereby producing the graph 𝐺 of Figure 3.7. Then 𝐺 has order 𝑝. Let
be the set of all extreme vertices and support vertex of 𝐺. By Theorem 2.4, every total detour monophonic set of G contains S. It is clear that for any
is not a total detour monophonic set of G. Since
is a total detour monophonic set of G, we have dm_{
t
} (G)= a .

Let
be the set of all extreme vertices and cut-vertices of G. By Theorems 1.3 and 1.4, every connected detour monophonic set of G contains S_{
1
} . It is clear that for any
is not a connected detour monophonic set of G. Let
. It is easily verified that
is a connected detour monophonic set of G and so dm_{
c
} (G)=b .

**Case 3**. 4 ≤ a = b < p. Let
be a path of order 3. Let H be the graph obtained from P_{
3
} by adding p - a new vertices
to 𝑃 3 and join each w_{i}(1 ≤ i ≤ p − a) to v_{
1
} and v_{
3
} . Also, add 𝑎−3 new vertices
to H and join each u_{
i
} (1 ≤ i ≤ a − 3) with v_{
2
} , thereby producing the graph G in Figure 3.8 of order p. Let
be the set of all extreme vertices and support vertex of G. By Theorems 2.4, 1.3 and 1.4 every total detour monophonic set and every connected detour monophonic set of G contains 𝑆. It is clear that for any
is not a total detour monophonic set of G. Since
is a connected detour monophonic set of G and also a total detour monophonic set of G, we have dm_{
c
} (G)= dm_{
t
} (G)= a .

**Case 4.** 4 ≤ a < b = p. Let
be a path of order b − a + 3. Add a − 3 new vertices
to P_{
b−a+3
} and join each v_{
i
} (1 ≤ i ≤ a −3) with u_{
b−a+3,
} thereby producing the graph G in Figure 3.9 of order p=b . Let
be the set of all extreme vertices and support vertices of G. By Theorem 2.4, every total detour monophonic set of G contains S. It is clear that S is the total detour monophonic set of G and so dm_{
t
} (G)= a.

It is clear that V(G) is the set of all extreme vertices and cut-vertices of G. Then by Theorems 1.3 and 1.4, V(G) is the connected detour monophonic set of G and so dm_{
c
} (G) = b = p.

**Theorem 3.6.** For positive integers a,b such that 4 ≤ a ≤ b with b ≤ 2a, there exists a connected graph G such that dm(G) = a and dm_{
t
} (G) = b.

**Proof. Case 1**. 4 ≤ a = b. Then the complete graph K_{
a
} has the desired properties.

**Case 2**. 4 ≤ a < b. Let b = a+k, where 1 ≤ k ≤ a. For k = 1, the star K_{
1,a
} has the desired properties. Now, let k ≥ 2. Let
be K-1 copies of C_{
4
} . Let H be the graph formed by identifying the vertices x_{
i
} from
and let x be the identified vertex. Let 𝐺 be the graph obtained from 𝐻 by adding a-k+1 new vertices
and joining each
with x. The graph G is shown in Figure 3.10. Let
be the set of all extreme vertices of G. By Theorem 1.1, every detour monophonic set of G contains S. It is clear that
is the detour monophonic set of G and so dm(G) = a.

Let
be the set of all extreme vertices and support vertex of G. By Theorem 2.4, every total detour monophonic set of G contains S’. Also, by Theorem 2.6, every total detour monophonic set of G contains at least one vertex from each
is a minimum total detour monophonic set of G so that dm_{
t
} (G)=b.