Monophonic Distance in Graphs

Santhakumaran, A. P.; Titus, P.

doi:10.1142/s1793830911001176

Cited by 30 publications

(21 citation statements)

References 26 publications

(30 reference statements)

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“…A similar result is given in the next theorem, and for a proof, one may refer to Ref. [45]. Theorem 2.18.…”

Section: Graph Theory -Advanced Algorithms and Applicationssupporting

confidence: 58%

“…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.…”

Section: Theorem 35 No Cut Vertex Of a Connected Graph G Belongs Tomentioning

confidence: 99%

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Monophonic Distance in Graphs

Titus¹,

Santhakumaran²

2018

Graph Theory - Advanced Algorithms and Applications

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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 distance 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 monophonic 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.Keywords: monophonic path, monophonic distance, detour monophonic number, upper detour monophonic number, forcing detour monophonic number, vertex detour monophonic number, upper vertex detour monophonic number, forcing vertex detour monophonic number Graph Theory -Advanced Algorithms and Applications 116The 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. ThenResult 2.4. Let u and v be any two vertices in a connected graph G. Then (i) d m (u, v) = 0 if and only if u = v. (ii) d m (u, v) = 1 if and only if uv is an edge of G. (iii) d m (u, v) = p − 1 if and only if G is the path with endvertices u and v.(iv) 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.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. Figure 1. The graph G in Example 2.2.Monophonic Distance in Graphs http://dx.doi.org/10.5772/intechopen.68668 117Graph Theory -Advanced Algorithms and Applications 132

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“…A similar result is given in the next theorem, and for a proof, one may refer to Ref. [45]. Theorem 2.18.…”

Section: Graph Theory -Advanced Algorithms and Applicationssupporting

confidence: 58%

Section: Theorem 35 No Cut Vertex Of a Connected Graph G Belongs Tomentioning

confidence: 99%

Monophonic Distance in Graphs

Titus¹,

Santhakumaran²

2018

Graph Theory - Advanced Algorithms and Applications

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“…It is shown in [7] 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. It can be extended so that the upper vertex monophonic number can be prescribed.…”

Section: Bounds and Realization Results For Mmentioning

confidence: 99%

The Upper Vertex Monophonic Number of a Graph

Titus¹,

Iyappan²

2016

Int. J. of Pure and Appl. Math.

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For any vertex x in a connected graph G of order p ≥ 2, a set S ⊆ V (G) is an x-monophonic set of G if each vertex v ∈ V (G) lies on an x − y monophonic path 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, denoted by mx(G). An x-monophonic set S is called a minimal x-monophonic set if no proper subset of S is an x-monophonic set. The upper x-monophonic number, denoted by m + x (G), is defined as the maximum cardinality of a minimal x-monophonic set of G. We determine bounds for it and find the same for some special classes of graphs.For any two positive integers a and b with 1 ≤ a ≤ b, there exists a connected graph G with mx(G) = a and m + x (G) = b for some vertex x in G. Also, it is shown that for any three positive integers a, b and n with a ≥ 2 and a ≤ n ≤ b, there exists a connected graph G with mx(G) = a, m + x (G) = b and a minimal x-monophonic set of cardinality n.

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Edge fixed monophonic number of a graph

Titus

Vanaja²

2017

Proyecciones (Antofagasta)

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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 ∈ 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 m xy (G). An xy-monophonic set of cardinality m xy (G) is called a m xy -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 m xy (G) = n for some edge xy in G.

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Monophonic Distance in Graphs

Cited by 30 publications

References 26 publications

Monophonic Distance in Graphs

Monophonic Distance in Graphs

The Upper Vertex Monophonic Number of a Graph

Edge fixed monophonic number of a graph

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