Monophonic Distance in Graphs Monophonic Distance in Graphs

For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distanc e d m ( u , v ) is the length of a longest u − v monophonic path in G . For any vertex v in G , the monophonic eccentricity of v is e m ( v ) = max { d m ( u , v ) : u ∈ V }. The subgraph induced by the vertices of G having minimum mono- phonic eccentricity is the monophonic center of G , and it is proved that every graph is the monophonic center of some graph. Also it is proved that the monophonic center of every connected graph G lies in some block of G. With regard to convexity, this monophonic distance is the basis of some detour monophonic parameters such as detour monophonic number, upper detour monophonic number, forcing detour monophonic number, etc. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph would be introduced and discussed. Various interesting results based on these parameters are also discussed in this chapter.


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 real-life 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.

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 : 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, 6 ), and so the triangle inequality is not satisfied.
The following result is an easy consequence of the respective definitions.   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.

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 The monophonic radius and the monophonic diameter of some standard graphs are given in Table 2. The next theorem follows from Proposition 2.3. Theorem 2.9. For a connected graph G, the following results hold:  Proof. (a) The result is proved by considering three cases.   Table 1. Monophonic eccentricities of the vertices of the graph G in Figure 1.

Graph Theory -Advanced Algorithms and Applications
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 The resulting graph G is shown in Figure 2.
. It follows that rad G = a, rad m G = b, and rad D G = c.
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 The resulting graph G is shown in Figure 3.

It is easily verified that
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.
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 with v a−2 and w i . The graph G is shown in Figure 5.

It is easily verified that
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 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. Proof. This result is proved by considering three cases.

and the subgraph induced by the monophonic peripheral vertices of G is the monophonic periphery
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 }. 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. 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]. 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 self-centered graphs.
The following problem is left open.

Problem 2.21. Characterize monophonic self-centered graphs.
Further results on monophonic distance in graphs can be found in Refs. [45,46].

Detour monophonic number
Throughout this section, by a u − v detour monophonic path, we mean a longest u − v monophonic path.  Monophonic Distance in Graphs http://dx.doi.org/10.5772/intechopen.68668 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 3.5. No cut vertex of a connected graph G belongs to any minimum detour monophonic set of G.
Since every end-block 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.

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 : 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].  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.   The next theorem is an interesting realization result, and for a proof, one may refer to Ref. [48].

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.   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.  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]. Further results on detour monophonic concepts in graphs can be found in Refs. [47][48][49][50].

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. 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.
The next two theorems are easy to prove.  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 x-detour 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.  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.  Table 3. Vertex detour monophonic numbers of the graph G in Figure 11. Since every minimum x-detour monophonic set is a minimal x-detour 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].     Table 4. Upper vertex detour monophonic numbers of the graph G in Figure 12. 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. 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]. There are useful applications of these concepts to security-based 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 real-life application of communication networks. The edge detour monophonic sets are discussed in Refs. [53][54][55].   Table 5. Forcing and upper forcing vertex detour monophonic numbers of the graph G in Figure 13.

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.