1. Introduction
In this chapter, we consider a finite connected graph G = (V(G), E(G)) having no loops and multiple edges. The order and size of G are denoted by p and q, respectively. Distance in graphs is a wide branch of graph theory having numerous scientific and reallife applications. There are many kinds of distances in graphs found in literature. For any two vertices u and v in G, the distance d(u, v) from u to v is defined as the length of a shortest u − v path in G. The eccentricity e(v) of a vertex v in G is the maximum distance from v to a vertex of G. The radius rad G of G is the minimum eccentricity among the vertices of G, while the diameter diam G of G is the maximum eccentricity among the vertices of G. The distance between two vertices is a fundamental concept in pure graph theory, and this distance is a metric on the vertex set of G. More results related to this distance are found in Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9]. This distance is used to study the central concepts like center, median, and centroid of a graph [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. With regard to convexity, this distance is the basis of some geodetic parameters such as geodetic number, connected geodetic number, upper geodetic number and forcing geodetic number [23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. The geodesic graphs, extremal graphs, distance regular graphs and distance transitive graphs are some important classes based on the distance in graphs [33, 34]. These concepts have interesting applications in location theory and convexity theory. 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.
The detour distance, which is defined to be the length of a longest path between two vertices of a graph, is also a metric on the vertex set of G [35, 36]. For any two vertices u and v in G, the detour distance D(u, v) from u to v is defined as the length of a longest u − v path in G. The detour eccentricity e_{D}(v) of a vertex v in G is the maximum detour distance from v to a vertex of G. The detour radius rad_{D}G of G is the minimum detour eccentricity among the vertices of G, while the detour diameter diam_{D}G of G is the maximum detour eccentricity among the vertices of G. With regard to detour convexity, the detour number of a graph was introduced and studied in Refs. [25, 37]. The detour concepts and colorings are widely used in the channel assignment problem in FM radio technology and also in certain molecular problems in theoretical chemistry.
The parameter geodetic (detour) number of a graph is global in the sense that there is exactly one geodetic (detour) number for a graph. The concept of geodetic (detour) sets and geodetic (detour) numbers by fixing a vertex of a graph was also introduced and discussed in Refs. [38, 39, 40, 41, 42]. With respect to each vertex of a graph, there is a geodetic (detour) number, and so there will be at most as many geodetic (detour) numbers as there are vertices in the graph.
2. Monophonic distance
Definition 2.1.A chord of a path u_{1}, u_{2},…, u_{n}in a connected graph 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. The length of a longest u − v monophonic path is called the monophonic distance from u to v, and it is denoted by d_{m}(u, v). A u − v monophonic path with its length equal to d_{m}(u, v) is known as a u − v monophonic.
Example 2.2. Consider the graph G given in Figure 1. It is easily verified that d(v_{1}, v_{4}) = 2, D(v_{1}, v_{4}) = 6, and d_{m}(v_{1}, v_{4}) = 4. Thus the monophonic distance is different from both the distance and the detour distance. The monophonic path P : v_{1}, v_{2}, v_{8}, v_{7}, v_{4} is v_{1}− v_{4} monophonic while the monophonic path Q : v_{1}, v_{3}, v_{4} is not v_{1}− v_{4} monophonic.
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 d_{m} is not a metric on V. For the graph G given in Figure 1, d_{m}(v_{4}, v_{6}) = 5, d_{m}(v_{4}, v_{5}) = 1 and d_{m}(v_{5}, v_{6}) = 1. Hence d_{m}(v_{4}, v_{6}) > d_{m}(v_{4}, v_{5}) + d_{m}(v_{5}, v_{6}), and so the triangle inequality is not satisfied.
The following result is an easy consequence of the respective definitions.
Proposition 2.3.Let u and v be any two vertices in a graph G of order p. then
Result 2.4. Let u and v be any two vertices in a connected graph G. Then
d_{m}(u, v) = 0 if and only if u = v.
d_{m}(u, v) = 1 if and only if uv is an edge of G.
d_{m}(u, v) = p − 1 if and only if G is the path with endvertices u and v.
d(u, v) = d_{m}(u, v) = D(u, v) if and only if G is a tree.
Definition 2.5.For any vertex v in a connected graph G, the monophonic eccentricity of v is e_{m}(v) = max {d_{m}(u, v) : u ∈ V}. A vertex u of G such that d_{m}(u, v) = e_{m}(v) is called a monophonic eccentric vertex of v. The monophonic radius and monophonic diameter of G are defined by rad_{m}G = min {e_{m}(v) : v ∈ V} and diam_{m}G = max {e_{m}(v) : v ∈ V}, respectively.
Example 2.6.Table 1 shows the monophonic distance between the vertices and also the monophonic eccentricities of vertices of the graph G given in Figure 1. It is to be noted that rad_{m}G = 3 and diam_{m}G = 5.
d_{m}(v_{i}, v_{j})  v_{1}  v_{2}  v_{3}  v_{4}  v_{5}  v_{6}  v_{7}  v_{8}  e_{m}(v) 

v_{1}  0  1  1  4  1  4  3  4  4 
v_{2}  1  0  4  3  1  5  4  1  5 
v_{3}  1  4  0  1  2  4  4  4  4 
v_{4}  4  3  1  0  1  5  1  4  5 
v_{5}  1  1  2  1  0  1  3  3  3 
v_{6}  4  5  4  5  1  0  1  1  5 
v_{7}  3  4  4  1  3  1  0  1  4 
v_{8}  4  1  4  4  3  1  1  0  4 
Remark 2.7. In any connected graph, the eccentricity of every two adjacent vertices differs by at most 1. However, this is not true in the case of monophonic distance. For the graph G given in Figure 1, e_{m}(v_{5}) = 3 and e_{m}(v_{6}) = 5.
Note 2.8. Any two vertices u and v in a tree T are connected by a unique path, and so d(u, v) = d_{m}(u, v) = D(u, v). Hence rad T = rad_{m}T = rad_{D}T and diam T = diam_{m}T = diam_{D}T. The monophonic radius and the monophonic diameter of some standard graphs are given in Table 2.
Graph G  K_{p}  C_{p}  W_{1,p−1} (p ≥ 4)  K_{1,p−1} (p ≥ 2)  K_{m,n} (m,n ≥ 2)  P_{p} 

rad_{m}G  1  p − 2  1  1  2 

diam_{m}G  1  p − 2  p − 3  2  2  p − 1 
The next theorem follows from Proposition 2.3.
Theorem 2.9.For a connected graph G, the following results hold:
e(v) ≤ e_{m}(v) ≤ e_{D}(v) for any vertex v in G.
rad G ≤ rad_{m}G ≤ rad_{D}G.
diam G ≤ diam_{m}G ≤ diam_{D}G.
Theorem 2.10. (a) If a, band c are integers with 3 ≤ a ≤ b ≤ c, then there exists a connected graph G such that rad G = a, rad_{m}G = band rad_{D}G = c.
(b) If a, band c are integers with 5 ≤ a ≤ b ≤ c, then there exists a connected graph G such that diam G = a, diam_{m}G = band diam_{D}G = c.
Proof. (a) The result is proved by considering three cases.
Case (i) 3 ≤ a = b = c. Consider G = P_{2a + 1}, the path of order 2a + 1. It is clear that rad G = rad_{m} G = rad_{D}G = a.
Case (ii) 3 ≤ a ≤ b < c. Let F_{1} : u_{1}, u_{2},…, u_{a−1} and F_{2} : v_{1}, v_{2},…, v_{a−1} be two copies of the path P_{a−1} of order a − 1. Let F_{3} : w_{1}, w_{2},…, w_{b−a+3} and F_{4} : z_{1}, z_{2},…, z_{b−a+3} be two copies of the path P_{b−a+3} of order b−a+3, and F_{5} = K_{c−b+1} the complete graph of order c − b + 1 with V(F_{5}) = {x_{1}, x_{2},…, x_{c−b+1}}. We construct the graph G as follows: (i) identify the vertices x_{1} in F_{5} and w_{1} in F_{3}; also identify the vertices x_{c−b+1} in F_{5} and z_{1} in F_{4}; (ii) identify the vertices w_{b−a+3} in F_{3} and u_{2} in F_{1}, and identify the vertices z_{b−a+3} in F_{4} and v_{2} in F_{2}; and (iii) join each vertex w_{i} (1 ≤ i ≤ b−a+2) in F_{3} and u_{1} in F_{1}, and join each vertex z_{i} (1 ≤ i ≤ b−a+2) in F_{4} and v_{1} in F_{2}. The resulting graph G is shown in Figure 2. It is easily verified that e(v) = a if v ∈ V(F_{5}); e(v) > a if v ∈ V(G) − V(F_{5}), e_{m}(v) = b if v ∈ V(F_{5}); e_{m}(v) > b if v ∈ V(G) − V(F_{5}) and e_{D}(v) = c if v ∈ V(F_{5}); and e_{D}(v) > c if v ∈ V(G) − V(F_{5}). It follows that rad G = a, rad_{m}G = b, and rad_{D}G = c.
Case (iii) 3 ≤ a < b = c. Let E_{1} : v_{1}, v_{2},…, v_{2a+1} be a path of order 2a + 1. Let E_{2} : u_{1}, u_{2},…, u_{b−a+3} and E_{3}: w_{1}, w_{2},…, w_{b−a+3} be two copies of the path P_{b−a+3} of order b − a + 3, and let E_{i} (4 ≤ i ≤ 2(b − a) + 3) be 2(b − a) copies of K_{1}. We construct the graph G as follows: (i) identify the vertices v_{a+1} in E_{1}, u_{1} in E_{2}, and w_{1} in E_{3}; (ii) identify the vertices v_{a−1} in E_{1} and u_{b−a+3} in E_{2}, and identify the vertices v_{a+3} in E_{1} and w_{b−a+3} in E_{3}; and (iii) join each E_{i} (4 ≤ i ≤ b − a + 3) with v_{a+1} in E_{1} and u_{i−1} in E_{2}, and join each E_{i} (b − a + 4 ≤ i ≤ 2(b − a) + 3) with v_{a+1} in E_{1} and w_{i−b+a−1} in E_{3}. The resulting graph G is shown in Figure 3.
It is easily verified that e(v_{a+1}) = a; e(v) > a if v ∈ V(G) − {v_{a+1}}; e_{m}(v_{a+1}) = b; e_{m}(v) > b if v ∈ V(G) − {v_{a+1}}, and e_{D}(v_{a+1}) = c; and e_{D}(v) > c if v ∈ V(G) − {v_{a+1}}. It follows that rad G = a, rad_{m}G = b, and rad_{D}G = c.
(b) This result is also proved by considering three cases.
Case (i) 5 ≤ a = b = c. Let G be a path of order a+1. Then diam G = diam_{m}G = diam_{D}G = a.
Case (ii) 5 ≤ a ≤ b < c. Let F_{1} : u_{1}, u_{2},…, u_{a−1} be the path P_{a−1} of order a − 1; F_{2} : w_{1}, w_{2},…, w_{b−a+3} be the path P_{b−a+3} of order b − a + 3; and F_{3} = K_{c−b+1} be the complete graph of order c − b +1 with V (F_{3}) = {x_{1}, x_{2},…, x_{c−b+1}}. We construct the graph G as follows: (i) identify the vertices x_{1} in F_{3} and w_{1} in F_{2}, and identify the vertices w_{b−a+3} in F_{2} and u_{2} in F_{1}, and (ii) join each vertex w_{i} (1 ≤ i ≤ b − a + 2) in F_{2} and u_{1} in F_{1}. The resulting graph G is shown in Figure 4. It is easily verified that e(v) = a if v ∈ (V(F_{3}) − {x_{1}}) ∪ {u_{a−1}}; e(v) < a if v ∈ V(F_{2}) ∪ (V(F_{1}) − {u_{a−1}}), and e_{m}(v) = b if v ∈ (V(F_{3}) − {x_{1}}) ∪ {u_{a−1}}; e_{m}(v) < b if v ∈ V(F_{2}) ∪ (V(F_{1}) − {u_{a−1}}), and e_{D}(v) = c if v ∈ (V(F_{3}) − {x_{1}}) ∪ {u_{a−1}}; and e_{D}(v) < c if v ∈ V(F_{2}) ∪ (V(F_{1}) − {u_{a−1}}). It follows that diam G = a, diam_{m}G = b and diam_{D}G = c.
Case (iii) 5 ≤ a < b = c. Let E_{1} : v_{1}, v_{2},…, v_{a+1} be a path of order a + 1; E_{2} : w_{1}, w_{2},…, w_{b−a+3} be another path of order b − a + 3; and E_{i} (3 ≤ i ≤ b − a + 2) be b − a copies of K_{1}. Let G be the graph obtained from E_{i} for i = 1, 2,…, b − a + 2 by identifying the vertices v_{a−2} and v_{a} of E_{1} with w_{1} and w_{b−a+3} of E_{2}, respectively, and joining each E_{i} (3 ≤ i ≤ b − a + 2) with v_{a−2} and w_{i}. The graph G is shown in Figure 5.
It is easily verified that e(v) = a if v ∈ {v_{1}, v_{a+1}}; e(v) ≤ a if v ∈ V(G) − {v_{1}, v_{a+1}}, and e_{m}(v) = b if v ∈ {v_{1}, v_{a+1}}; e_{m}(v) ≤ b if v ∈ V(G) − {v_{1}, v_{a+1}}, and e_{D}(v) = c if v ∈ {v_{1}, v_{a+1}}; and e_{D}(v) ≤ c if v ∈ V(G) − {v_{1}, v_{a+1}}. It follows that rad G = a, rad_{m}G = b and rad_{D}G = c.
For any connected graph G, the inequalities rad G ≤ diam G ≤ 2 rad G and rad_{D}G ≤ diam_{D}G ≤ 2 rad_{D}G hold. However, this is not true in the case of monophonic radius and monophonic diameter. For example, when the graph G is the wheel W_{1,p−1} (p ≥ 6), it is easily seen that rad_{m}G = 1 and diam_{m}G = p − 3 ≥ 3 so that diam_{m}G > 2 rad_{m}G.
It is proved in Ref. [6] that if a and b are any two positive integers such that a ≤ b ≤ 2a, then there is a connected graph G with rad G = a and diam G = b. Also, it is proved in Ref. [35] that if a and b are any two positive integers such that a ≤ b ≤ 2a, then there is a connected graph G with rad_{D}G = a and diam_{D}G = b.
Now, the following theorem gives a realization result for rad_{m}G ≤ diam_{m}G.
Theorem 2.11.If a and b are positive integers with a ≤ b, then there exists a connected graph G such that rad_{m}G = a and diam_{m}G = b.
Proof. This result is proved by considering three cases.
Case (i)a = b ≥ 1. Let G be the cycle C_{a+2}. Then rad_{m}G = a and diam_{m}G = b.
Case (ii)a < b ≤ 2a. Let C_{1} : u_{1}, u_{2},…, u_{a+2}, u_{1} be a cycle of order a + 2 and C_{2} : v_{1}, v_{2},…, v_{b−a+2}, v_{1} be a cycle of order b − a + 2. Let G be the graph obtained by identifying the vertex u_{1} of C_{1} and v_{1} of C_{2}. Since b ≤ 2a, it follows that b − a + 2 ≤ a + 2. It is clear that d_{m}(u_{1}, x) ≤ a for any x in G and d_{m}(u_{1}, u_{a+1}) = a. Therefore, e_{m}(u_{1}) = a. Also, it is clear that there is no vertex x with e_{m}(x) < a and so rad_{m}G = a. It is clear that d_{m}(u_{3}, v_{3}) = b and d_{m}(u_{3}, x) ≤ b for any vertex x in G and so e_{m}(u_{3}) = b. Also, it is easy to see that e_{m}(x) ≤ b for every vertex x in G so that diam_{m}G = b.
Case (iii)b > 2a. Let G be the graph obtained by identifying the central vertex of the wheel W = K_{1} + C_{b+2} (b ≥ 2) and an endvertex of the path P_{2a}. Since b > 2a, e_{m}(x) = b for any vertex x ∈ V(C_{b+2}). Also, a ≤ e_{m}(x) ≤ 2a for any vertex x ∈ V(P_{2a}) and e_{m}(v_{a}) = a. Hence rad_{m}G = a and diam_{m}G = b.
2.1. Monophonic center and monophonic periphery
Definition 2.12.A vertex v in a connected graph G is called a monophonic central vertex if e_{m}(v) = rad_{m}G, and the subgraph induced by the monophonic central vertices of G is the monophonic center C_{m}(G) of G. A vertex v in G is called a monophonic peripheral vertex if e_{m}(v) = diam_{m}G, and the subgraph induced by the monophonic peripheral vertices of G is the monophonic periphery P_{m}(G) of G.
Example 2.13. Consider the graph G given in Figure 1. It is easily verified that v_{5} is the monophonic central vertex and v_{2}, v_{4}, and v_{6} are the monophonic peripheral vertices of G.
Remark 2.14. The monophonic center of a connected graph need not be connected. For the graph G given in Figure 6, C_{m}(G) = {v_{3}, v_{6}}.
Theorem 2.15.Every graph is the monophonic center of some connected graph.
Proof. Let G be a graph. We show that G is the monophonic center of some graph. Let l = d_{m} be the monophonic diameter of G. Let P : u_{1}, u_{2},…, u_{l} and Q : v_{1}, v_{2},…, v_{l} be two copies of the path P_{l}. The required graph H given in Figure 7 is got from G, P, and Q by joining each vertex of G with u_{1} in P and v_{1} in Q. Then e_{mH}(x) = d_{m} for each vertex x in G and d_{m} + 1 ≤ e_{mH}(x) ≤ 2 d_{m} for each vertex x not in G. Therefore, V(G) is the set of monophonic central vertices of H and so C_{m}(H) = G.
More specifically, it is proved in Ref. [43] that the center of every connected graph G lies in a single block of G. Also, it is proved in Ref. [35] that the detour center of every connected graph G lies in a single block of G. The same result is true for the monophonic center also, as proved in the following theorem.
Theorem 2.16.The monophonic center C_{m}(G) of every connected graph G is a subgraph of some block of G.
Proof. Suppose that there is a connected graph G such that its monophonic center C_{m}(G) is not a subgraph of a single block of G. Then G has a cut vertex v such that G − v contains two components H_{1} and H_{2}, each containing vertices of C_{m}(G). Let u be a vertex of G such that e_{m}(v) = d_{m}(u,v), and let P_{1} be a u − v longest monophonic path in G. Then at least one of H_{1} and H_{2} contains no vertices of P_{1}, say H_{2} contains no vertex of P_{1}. Now, take a vertex w in C_{m}(G) that belongs to H_{2}, and let P_{2} be a v − w longest monophonic path in G. Since v is a cut vertex, P_{1} followed by P_{2} gives a u − w longest monophonic path with its length greater than that of P_{1}. This gives e_{m}(w) > e_{m}(v) so that w is not a monophonic central vertex of G, which is a contradiction.
Corollary 2.17.For any tree, the monophonic center is isomorphic to K_{1}or K_{2}.
It is proved in Ref. [44] that a nontrivial graph G is the periphery of some connected graph if and only if every vertex of G has eccentricity 1 or no vertex of G has eccentricity 1. Also, it is proved in Ref. [35] that a connected graph G of order p ≥ 3 and radius 1 is the detour periphery of some connected graph if and only if G is Hamiltonian. A similar result is given in the next theorem, and for a proof, one may refer to Ref. [45].
Theorem 2.18.A nontrivial graph G is the monophonic periphery of some connected graph if and only if every vertex of G has monophonic eccentricity 1 or no vertex of G has monophonic eccentricity 1.
Definition 2.19.A connected graph G is monophonic selfcentered if rad_{m}G = diam_{m}G, that is, if G is its own monophonic center.
Example 2.20. The complete graph K_{n}, the cycle C_{n}, and the complete bipartite graph K_{m,n} (m, n ≥ 2) are monophonic selfcentered graphs.
The following problem is left open.
Problem 2.21.Characterize monophonic selfcentered graphs.
Further results on monophonic distance in graphs can be found in Refs. [45, 46].
3. Detour monophonic number
Throughout this section, by a u − v detour monophonic path, we mean a longest u − v monophonic path.
Definition 3.1.A set S of vertices of a connected graph G is called a detour monophonic set if every vertex of G lies on a u − v detour monophonic path for some u, v ∈ S. The detour monophonic number of G is defined as the minimum cardinality of a detour monophonic set of G and is denoted by dm(G).
Example 3.2. For the graph G given in Figure 8, S_{1} = {v_{1}, v_{2}, v_{3}}, S_{2} = {v_{2}, v_{3}, v_{4}}, S_{3} = {v_{5}, v_{6}, v_{2}}, S_{4} = {v_{5}, v_{6}, v_{3}}, S_{5} = {v_{1}, v_{3}, v_{5}}, S_{6} = {v_{1}, v_{3}, v_{6}}, S_{7} = {v_{2}, v_{4}, v_{5}}, and S_{8} = {v_{2}, v_{4}, v_{6}} are the minimum detour monophonic sets of G and so dm(G) = 3.
If a vertex belongs to every minimum detour monophonic set of G, then it is called a detour monophonic vertex of G. If S is the unique minimum detour monophonic set of G, then S is the set of all detour monophonic vertices of G. In the next theorem, we show that there are certain vertices in a nontrivial connected graph G that are detour monophonic vertices of G.
Theorem 3.3.Every detour monophonic set of a connected graph G contains all its extreme vertices. Moreover, if the set of all extreme vertices S of G is a detour monophonic set of G, then S is the unique minimum detour monophonic set of G.
Proof. Let v be an extreme vertex and let S be a detour monophonic set of G. If v is not an element of S, then there exist two elements x and y in S such that v is an internal vertex of an x − y detour monophonic path, say P. Let u and w be the vertices on P adjacent to v. Then u and w are not adjacent and so v is not an extreme vertex of G, which is a contradiction. Therefore v belongs to every detour monophonic set of G. Thus, if S is the set of all extreme vertices of G, then dm(G) ≥ S. On the other hand, if S is a detour monophonic set of G, then dm(G) ≤ S. Therefore dm(G) = S and S is the unique minimum detour monophonic set of G.
The following two theorems are easy to prove.
Theorem 3.4.Let G be a connected graph with a cut vertex vand let S be a detour monophonic set of G. Then every component of G − v contains an element of S.
Theorem 3.5.No cut vertex of a connected graph G belongs to any minimum detour monophonic set of G.
Since every endblock B is a branch of G at some cut vertex, it follows Theorem 3.4 and Theorem 3.5 that every minimum detour monophonic set of G contains at least one vertex from B that is not a cut vertex. Thus the following corollaries are consequences of Theorems 3.4 and 3.5.
Corollary 3.6.If G is a connected graph with k ≥ 2 endblocks, then dm(G) ≥ k.
Corollary 3.7.If k is the maximum number of blocks to which a vertex in a graph G belongs, then dm(G) ≥ k.
Theorem 3.8.For any connected graph G, 2 ≤ dm(G) ≤ p.
Theorem 3.9.For any connected graph G, dm(G) = p if and only if G is complete.
Proof. Let dm(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 detour monophonic path from u to v, say P, with length at least 2. Clearly, (V(G) − V(P)) ∪ {u, v} is a detour monophonic set of G and hence dm(G) ≤ p − 1, which is a contradiction. Conversely, if G is complete, then by Theorem 3.3, dm(G) = p.
Theorem 3.10.If G is a nontrivial connected graph of order p and monophonic diameter d, then dm(G) ≤ p − d + 1.
Proof. Let x, y ∈ V(G) such that G contains an x − y detour monophonic path P of length diam_{m}G = d. Let S = (V(G)−V(P)) ∪ {x, y}. Since S is a detour monophonic set of G, it follows that dm(G) ≤ S ≤ p − d + 1.
Theorem 3.11.For every nontrivial tree T of order p and monophonic diameter d, dm(T) = p − d + 1 if and only if T is a caterpillar.
Proof. Let T be any nontrivial tree. Let P : u = v_{0}, v_{1},…, v_{d} be a monophonic diametral path. Let k be the number of endvertices of T and let l be the number of internal vertices of T other than v_{1}, v_{2},…, v_{d−1}. Then d − 1 + l + k = p. By Theorem 3.3 and Theorem 3.5, dm(T) = k and so dm(T) = p − d − l + 1. Hence dm(T) = p − d + 1 if and only if l = 0, if and only if all the internal vertices of T lie on the monophonic diametral path P, and if and only if T is a caterpillar.
It is known that rad_{m}G ≤ diam_{m}G for a connected graph G. It is proved in Ref. [45] that if a and b are any two positive integers such that a ≤ b, then there is a connected graph G with rad_{m}G = a and diam_{m}G = b. The same result can also be extended so that the detour monophonic number can be prescribed when rad_{m}G < diam_{m}G, and for a proof, one may refer to Ref. [47].
Theorem 3.12.For positive integers r, dand n ≥ 4 with r < d, there exists a connected graph G with rad_{m}G = r, diam_{m}G = dand dm(G) = n.
Problem 3.13. For any three positive integers r, d and n ≥ 4 with r = d, does there exist a connected graph G with rad_{m}G = r, diam_{m}G = d and dm(G) = n?
3.1. Upper detour monophonic number
Definition 3.14.A detour monophonic set S of a connected graph G is called a minimal detour monophonic set if no proper subset of S is a detour monophonic set of G. The maximum cardinality of a minimal detour monophonic set of G is the upper detour monophonic number of G, denoted by dm^{+}(G).
Example 3.15. Consider the graph G given in Figure 8. The minimal detour monophonic sets are S_{1} = {v_{1}, v_{2}, v_{3}}, S_{2} = {v_{2}, v_{3}, v_{4}}, S_{3} = {v_{5}, v_{6}, v_{2}}, S_{4} = {v_{5}, v_{6}, v_{3}}, S_{5} = {v_{1}, v_{3}, v_{5}}, S_{6} = {v_{1}, v_{3}, v_{6}}, S_{7} = {v_{2}, v_{4}, v_{5}}, S_{8} = {v_{2}, v_{4}, v_{6}} and S_{9} = {v_{1}, v_{4}, v_{5}, v_{6}}. For this graph, the upper detour monophonic number is 4, and the detour monophonic number is 3.
Note 3.16. Every minimum detour monophonic set is a minimal detour monophonic set, but the converse is not true. For the graph G given in Figure 8, S_{9} is a minimal detour monophonic set, but it is not a minimum detour monophonic set of G.
The following three theorems are easy to prove.
Theorem 3.17.For any connected graph G, 2 ≤ dm(G) ≤ dm^{+}(G) ≤ p.
Theorem 3.18.For a connected graph G, dm(G) = p if and only if dm^{+}(G) = p.
Theorem 3.19.If G is a connected graph of order p with dm(G) = p − 1, then dm^{+}(G) = p − 1.
The next theorem is an interesting realization result, and for a proof, one may refer to Ref. [48].
Theorem 3.20.For any three positive integers a, b and n with 2 ≤ a ≤ n ≤ b, there is a connected graph G with dm(G) = a, dm^{+}(G) = b and a minimal detour monophonic set of cardinality n.
3.2. Forcing detour monophonic number
A connected graph G may contain more than one minimum detour monophonic sets. For example, the graph G given in Figure 8 contains eight minimum detour monophonic sets. For each minimum detour monophonic set S in G, there is always some subset T of S that uniquely determines S as the minimum detour monophonic set containing T. Such sets are called “forcing detour monophonic subsets” and these sets are discussed in this section.
Definition 3.21.Let S be a minimum detour monophonic set of a connected graph G. A subset S′ of S is a forcing detour monophonic subset for S if S is the unique minimum detour monophonic set that contains S′. A forcing detour monophonic subset for S of minimum cardinality is a minimum forcing detour monophonic subset of S. The cardinality of a minimum forcing detour monophonic subset of S is the forcing detour monophonic number fdm(S) in G. The forcing detour monophonic number of G is fdm(G) = min {fdm(S)}, where the minimum is taken over all minimum detour monophonic sets S in G.
Example 3.22.For the graph G given inFigure 9, S_{1} = {z, w, v}, S_{2} = {z, w, u} and S_{3} = {z, w, x} are the minimum detour monophonic sets of G. It is clear that fdm(S_{1}) = 1, fdm(S_{2}) = 1 and fdm(S_{3}) = 1 so that fdm(G) = 1. For the graph G given inFigure 10, S = {y, v} is the unique minimum detour monophonic set of G and so fdm(G) = 0.
The next theorem follows immediately from the definitions of the detour monophonic number and the forcing detour monophonic number of a graph G.
Theorem 3.23.For a connected graph G, 0 ≤ fdm(G) ≤ dm(G) ≤ p.
The following theorem characterizes graphs G for which fdm(G) = 0, fdm(G) = 1 and fdm(G) = dm(G). The proof is an easy consequence of the definitions of the detour monophonic number and the forcing detour monophonic number.
Theorem 3.24.Let G be a connected graph. Then
fdm(G) = 0 if and only if G contains exactly one minimum detour monophonic set.
fdm(G) = 1 if and only if G contains two or more minimum detour monophonic sets, one of which is a unique minimum detour monophonic set that contains one of its elements.
fdm(G) = dm(G) if and only if no minimum detour monophonic set of G is the unique minimum detour monophonic set that contains any of its proper subsets.
The next theorem gives a realization result for the parameters fdm(G) and dm(G), and for a proof, the reader may refer to Ref. [49].
Theorem 3.25.For every pair a, b of positive integers with 0 ≤ a < b and b ≥ 2, there exists a connected graph G such that fdm(G) = a and dm(G) = b.
Further results on detour monophonic concepts in graphs can be found in Refs. [47, 48, 49, 50].
4. Vertex detour monophonic number
The parameter detour monophonic number of a graph is global in the sense that there is exactly one detour monophonic number for a graph. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph was also introduced and discussed in this section. With respect to each vertex of a graph, there is a detour monophonic number, and so there will be at most as many detour monophonic numbers as there are vertices in the graph.
Definition 4.1.For any vertex x in a connected graph G, a set S_{x}of vertices in G is called an xdetour monophonic set if every vertex of G lies on an x − y detour monophonic path in G for some y in S_{x}. The xdetour monophonic number of G, denoted by dm_{x}(G), is defined to be the minimum cardinality of an xdetour monophonic set of G. An xdetour monophonic set of cardinality dm_{x}(G) is called a dm_{x}set of G.
It is easy to observe that for any vertex x in G, x does not belong to any dm_{x}set of G.
Example 4.2. For the graph G given in Figure 11, the minimum vertex detour monophonic sets and the vertex detour monophonic numbers are given in Table 3.
Vertex  Minimum vertex detour monophonic sets  Vertex detour monophonic number 

t  {z,w}  2 
y  {w,z,t}, {w,z,u}  3 
z  {u,w}, {w,y}  2 
u  {w,z,y}  3 
v  {w,t,z}, {w,u,z}  3 
w  {t,z}, {z,u}  2 
The next two theorems are easy to prove.
Theorem 4.3.For any vertex x in a connected graph G, the following results hold.
Every dm_{x}set of G contains all its extreme vertices other than the vertex x (whether x is extreme vertex or not).
No dm_{x}set of G contains a cut vertex of G.
Theorem 4.4.For any vertex x in a connected graph G of order p, 1 ≤ dm_{x}(G) ≤ p − 1.
Theorem 4.5.For any vertex x in a connected graph G of order p, dm_{x}(G) = p − 1 if and only if deg x = p − 1.
Proof. Let x be any vertex in a connected graph G of order p. Let dm_{x}(G) = p − 1. If deg x < p−1, then there is a vertex u in G that is not adjacent to x. Since G is connected, there is a detour monophonic path from x to u, say P, with length greater than or equal to 2. Then (V(G)−V(P)) ∪ {u} is an xdetour monophonic set of G so that dm_{x}(G) ≤ p − 2, which is a contradiction. Conversely, let deg x = p − 1. Hence x is adjacent to all other vertices of G. This shows that all these vertices form the dm_{x}set of G and so dm_{x}(G) = p − 1.
Corollary 4.6.A graph G is complete if and only if dm_{x}(G) = p − 1 for every vertex x in G.
4.1. Upper vertex detour monophonic number
Definition 4.7.Let x be any vertex of a connected graph G. An xdetour monophonic set S_{x}is called a minimal xdetour monophonic set if no proper subset of S_{x}is an xdetour monophonic set. The upper xdetour monophonic number is the maximum cardinality of a minimal xdetour monophonic set of G and is denoted by dm_{x}^{+}(G).
Example 4.8. For the graph G given in Figure 12, the minimum vertex detour monophonic sets the vertex detour monophonic numbers, the minimal vertex detour monophonic sets and the upper vertex detour monophonic numbers are given in Table 4.
Since every minimum xdetour monophonic set is a minimal xdetour monophonic set, we have 1 ≤ dm_{x}(G) ≤ dm_{x}^{+}(G) ≤ p − 1. In view of this, we have the following theorems, and for proofs one may refer to Ref. [51].
Theorem 4.9.Let x be any vertex in a connected graph G of order p ≥ 3. If dm_{x}(G) = 1, then dm_{x}^{+}(G) ≤ p − 2.
Theorem 4.10.Let x be any vertex in a connected graph G. Then dm_{x}(G) = p − 1 if and only if dm_{x}^{+}(G) = p − 1.
Theorem 4.11.For any four integers j, k, l and p with 2 ≤ j ≤ k ≤ l ≤ p − 7, there exists a connected graph G of order p with dm_{x}(G) = j, dm_{x}^{+}(G) = l and a minimal xdetour monophonic set of cardinality k.
4.2. Forcing vertex detour monophonic number
Definition 4.12.Let x be any vertex of a connected graph Gand let S_{x}be a minimum xdetour monophonic set of G. A subset S′ of S_{x}is an xforcing subset for S_{x}if S_{x}is the unique minimum xdetour monophonic set that contains S′. An xforcing subset for S_{x}of minimum cardinality is a minimum xforcing subset of S_{x}. The cardinality of a minimum xforcing subset of S_{x}is the forcing xdetour monophonic number fdm_{x}(S_{x}) in G. The forcing xdetour monophonic number of G is fdm_{x}(G) = min { fdm_{x}(S_{x})}, where the minimum is taken over all minimum xdetour monophonic sets S_{x}in G.
Definition 4.13.Let x be any vertex of a connected graph G. The upper forcing xdetour monophonic number, fdm_{x}^{+}(G), of G is the maximum forcing xdetour monophonic number among all minimum xdetour monophonic sets of G.
Example 4.14. For the graph G given in Figure 13, the minimum vertex detour monophonic sets, the vertex detour monophonic numbers, the forcing vertex detour monophonic sets, the forcing vertex detour monophonic numbers and the upper forcing vertex detour monophonic numbers are given in Table 5.
Theorem 4.15.For any vertex x in a connected graph G, 0 ≤ fdm_{x}(G) ≤ fdm_{x}^{+}(G) ≤ dm_{x}(G).
The following theorem gives a realization result for the parameters fdm_{x}(G), fdm_{x}^{+}(G), dm_{x}(G), and for a proof, one may refer to Ref. [52].
Theorem 4.16.For any three integers r, s and t with 2 ≤ r ≤ s ≤ t with 2r − s ≥ 0, there exists a connected graph G with fdm_{x}(G) = r, fdm_{x}^{+}(G) = sand dm_{x}(G) = t for some vertex x in G.
There are useful applications of these concepts to securitybased communication network design. In the case of designing the channel for a communication network, although all the vertices are covered by the network when considering detour monophonic sets, some of the edges may be left out. This drawback is rectified in the case of edge detour monophonic sets so that considering edge detour monophonic sets is more advantageous to reallife application of communication networks. The edge detour monophonic sets are discussed in Refs. [53, 54, 55].
5. Conclusion
In this chapter, the new distance known as monophonic distance in a graph is introduced, and its properties are studied. Its relationship with the usual distance and detour distance is discussed. Various realization theorems are proved with regard to the radius (diameter), monophonic radius (monophonic diameter) and detour radius (detour diameter). Results regarding monophonic center and monophonic periphery of a graph are presented. Further, the concept of a detour monophonic set in a graph is introduced and its various properties are presented. Consequently, the parameters, viz., detour monophonic number, upper detour monophonic number and forcing detour monophonic number of a graph are introduced and studied. In a similar way, the vertex detour monophonic number, the upper vertex detour monophonic number and the forcing vertex detour monophonic number of a graph are introduced and studied. Many interesting characterization theorems and also realization theorems with regard to all these parameters are presented. The results presented in this chapter would help the researchers in graph theory to develop new results and applications to various branches of science.