On the Monophonic Number of a Graph

Santhakumaran, A. P.; Titus, P.; Ganesamoorthy, K.

doi:10.14317/jami.2014.255

Cited by 22 publications

(16 citation statements)

References 3 publications

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“…16 Let G and H be arbitrary connected non-trivial graphs where H is not a complete graph and G is not isomorphic to K 2 . Suppose that the following conditions are satisfied:1. tn(G) = 2 = |Ext(G)| with Ext(G) = {u, v}; 2. tn(H) > 2;…”

mentioning

confidence: 99%

Toll number of the Cartesian and the lexicographic product of graphs

Gologranc

Repolusk

2017

Discrete Mathematics

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Toll convexity is a variation of the so-called interval convexity. A tolled walk T between u and v in G is a walk of the form T : u, w 1 , . . . , w k , v, where k ≥ 1, in which w 1 is the only neighbor of u in T and w k is the only neighbor of v in T . As in geodesic or monophonic convexity, toll interval betweenx lies on a tolled walk between u and v}. A set of vertices S is toll convex, ifFirst part of the paper reinvestigates the characterization of convex sets in the Cartesian product of graphs. Toll number and toll hull number of the Cartesian product of two arbitrary graphs is proven to be 2. The second part deals with the lexicographic product of graphs. It is shown that if H is not isomorphic to a complete graph, tn(G • H) ≤ 3 · tn(G). We give some necessary and sufficient conditions for tn(G • H) = 3 · tn(G). Moreover, if G has at least two extreme vertices, a complete characterization is given. Also graphs with tn(G • H) = 2 are characterized -this is the case iff G has an universal vertex and tn(H) = 2. Finally, the formula for tn(G • H) is given -it is described in terms of the so-called toll-dominating triples.

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confidence: 99%

Toll number of the Cartesian and the lexicographic product of graphs

Gologranc

Repolusk

2017

Discrete Mathematics

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“…The monophonic diameter diam m (G) of G is diam m (G) = max {e m (a) : a ∈ V (G)}. A set S of points of a graph G is a monophonic set of G if each point a of G lies on an s − t monophonic path in G for some s,t ∈ S. The minimum cardinality of a monophonic set S is the monophonic number of G and is denoted by m(G), Santhakumaran [8]. A point a is a extreme point of a graph G if the subgraph induced by its neighborhood is complete.…”

Section: Introductionmentioning

confidence: 99%

Monophonic number of a graph with respect to monophonic radius

Kinsley¹,

Shanthi²

2020

Malaya J. Mat.

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Let G be a connected graph of order at least two. We study about the monophonic sets and define a new set called monophonic radius-distance monophonic set of G. The monophonic radius-distance monophonic set of different kinds of graphs are found and verified. A new parameter monophonic radius-distance monophonic number is introduced and some bounds are found for several graphs. A relation between monophonic number and the monophonic radius-distance monophonic number of G is found. A main result, for every two integers c and d such that 2 ≤ c ≤ d, there is a graph with c as the monophonic number and d as the monophonic radius-distance monophonic number of G is proved.

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“…A set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x − y monophonic path for some x, y ∈ S. The minimum cardinality of a monophonic set of G is the monophonic number of G and is denoted by m(G). The monophonic number of a graph was studied and discussed in [8]. A set S of vertices of G is called a double monophonic set of G if for each pair of vertices x, y in G there exist vertices u, v in S such that x, y lie on a u − v monophonic path.…”

Section: Introductionmentioning

confidence: 99%

Restrained Double Monophonic Number of a Graph

Santhakumaran

Ganesamoorthy²

2019

UMJ

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For a connected graph \(G\) of order at least two, a double monophonic set \(S\) of a graph \(G\) is a restrained double monophonic set if either \(S=V\) or the subgraph induced by \(V-S\) has no isolated vertices. The minimum cardinality of a restrained double monophonic set of \(G\) is the restrained double monophonic number of \(G\) and is denoted by \(dm_{r}(G)\). The restrained double monophonic number of certain classes graphs are determined. It is shown that for any integers \(a,\, b,\, c\) with \(3 \leq a \leq b \leq c\), there is a connected graph \(G\) with \(m(G) = a\), \(m_r(G) = b\) and \(dm_{r}(G) = c\), where \(m(G)\) is the monophonic number and \(m_r(G)\) is the restrained monophonic number of a graph \(G\).

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On the Monophonic Number of a Graph

Cited by 22 publications

References 3 publications

Toll number of the Cartesian and the lexicographic product of graphs

Toll number of the Cartesian and the lexicographic product of graphs

Monophonic number of a graph with respect to monophonic radius

Restrained Double Monophonic Number of a Graph

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