The upper connected geodetic number and forcing connected geodetic number of a graph

Santhakumaran, A. P.; Titus, P.; John, J.

doi:10.1016/j.dam.2008.06.005

Cited by 17 publications

(10 citation statements)

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“…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].…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

Monophonic Distance in Graphs

Titus¹,

Santhakumaran²

2018

Graph Theory - Advanced Algorithms and Applications

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“…The connected geodetic number of graph was studied in [11]. The upper connected geodetic number and forcing connected geodetic number of a graph were studied in [12]. The edge geodetic number of a graph was studied by in [9].…”

Section: Introductionmentioning

confidence: 99%

On the edge-to-vertex geodetic number of a graph

Santhakumaran¹,

John²

2012

MMN

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Let G D .V; E/ be a connected graph with at least three vertices. For vertices u and v in G; the distance d.u; v/ is the length of a shortest u v path in G: A u v path of length d.u; v/ is called a u v geodesic. For subsets A and B of V; the distance d.A; B/; is defined as d.A; B/ D mi n fd.x; y/ W x 2 A; y 2 Bg. A u v path of length d.A; B/ is called an A B geodesic joining the sets A; B Â V; where u 2 A and v 2 B: A vertex x is said to lie on an A B geodesic if x is a vertex of an A B geodesic. A set S Â E is called an edge-to-vertex geodetic set if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S: The edge-to-vertex geodetic number g ev .G/ of G is the minimum cardinality of its edge-to-vertex geodetic sets and any edge-to-vertex geodetic set of cardinality g ev .G/ is an edge-to-vertex geodetic basis of G: Any edge-to-vertex geodetic basis is also called a g ev-set of G: It is shown that if G is a connected graph of size q and diameter d; then g ev .G/ Ä q d C 2: It is proved that, for a tree T with q 2; g ev .T / D q d C 2 if and only if T is a caterpillar. For positive integers r; d and l 2 with r Ä d Ä 2r; there exists a connected graph G with rad G D r; d iam G D d and g ev .G/ D l: Also graphs G for which g ev .G/ D q; q 1 or q 2 are characterized.

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“…The minimum cardinality of a connected geodetic set of G is the connected geodetic number G and is denoted by g c (G). A connected geodetic set of cardinality g c (G) is called a g c -set of G or a connected geodetic basis of G. The upper connected geodetic number and the forcing connected geodetic number of a graph was introduced and studied by Santhakumaran et al in [9]. A connected geodetic set S in a connected graph G is called a minimal connected geodetic set if no proper subset of S is a connected geodetic set of G. The upper connected geodetic number g + c (G) is the maximum cardinality of a minimal connected geodetic set of G.…”

Section: Introductionmentioning

confidence: 99%

The upper connected edge geodetic number of a graph

Santhakumaran

John²

2012

Filomat

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For a non-trivial connected graph G, a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected edge geodetic set of G is the connected edge geodetic number of G and is denoted by g1c(G). A connected edge geodetic set of cardinality g1c(G) is called a g1c-set of G or connected edge geodetic basis of G. A connected edge geodetic set S in a connected graph G is called a minimal connected edge geodetic set if no proper subset of S is a connected edge geodetic set of G. The upper connected edge geodetic number g + 1c (G) is the maximum cardinality of a minimal connected edge geodetic set of G. Graphs G of order p for which g1c(G) = g + 1c = p are characterized. For positive integers r,d and n ≥ d + 1 with r ≤ d ≤ 2r, there exists a connected graph of radius r, diameter d and upper connected edge geodetic number n. It is shown for any positive integers 2 ≤ a < b ≤ c, there exists a connected graph G such that g1(G) = a, g1c(G) = b and g + 1c (G) = c.

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The upper connected geodetic number and forcing connected geodetic number of a graph

Cited by 17 publications

References 4 publications

Monophonic Distance in Graphs

Monophonic Distance in Graphs

On the edge-to-vertex geodetic number of a graph

The upper connected edge geodetic number of a graph

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