Some Parameters on Neighborhood Number of A Graph

Chaluvaraju, B.

doi:10.1016/j.endm.2009.03.020

Cited by 4 publications

(4 citation statements)

References 2 publications

(3 reference statements)

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“…Moreover this result is also true given that V t jj is the neighborhood number of G 0 t (see [33,43]).…”

Section: Conflict Of Interestmentioning

confidence: 81%

“…A generalization of these results was presented by [42]. In the same year 2009, [43] explored the concept of neighborhood number and its relationship with other parameters including domination number. The common neighborhood number and an algorithm for constructing have been discussed at length and presented also in [30].…”

Section: Introductionmentioning

confidence: 88%

“…The common neighborhood number and an algorithm for constructing have been discussed at length and presented also in [30]. Just as with average connectivity, average degree and average distance invariants, the common neighborhood number is also used as a measure for reliability and stability of a graph [33,43].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Average Distance of Neighborhood Graphs and Its Applications

Mwakilama¹,

Ali²,

Chidzalo³

et al. 2022

Recent Applications in Graph Theory

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Graph invariants such as distance have a wide application in life, in particular when networks represent scenarios in form of either a bipartite or non-bipartite graph. Average distance μ of a graph G is one of the well-studied graph invariants. The graph invariants are often used in studying efficiency and stability of networks. However, the concept of average distance in a neighborhood graph G′ and its application has been less studied. In this chapter, we have studied properties of neighborhood graph and its invariants and deduced propositions and proofs to compare radius and average distance measures between G and G′. Our results show that if G is a connected bipartite graph and G′ its neighborhood, then radG1′≤radG and radG2′≤radG whenever G1′ and G2′ are components of G′. In addition, we showed that radG′≤radG for all r≥1 whenever G is a connected non-bipartite graph and G′ its neighborhood. Further, we also proved that if G is a connected graph and G′ its neighborhood, then and μG1′≤μG and μG2′≤μG whenever G1′ and G2′ are components of G′. In order to make our claims substantial and determine graphs for which the bounds are best possible, we performed some experiments in MATLAB software. Simulation results agree very well with the propositions and proofs. Finally, we have described how our results may be applied in socio-epidemiology and ecology and then concluded with other proposed further research questions.

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“…Moreover this result is also true given that V t jj is the neighborhood number of G 0 t (see [33,43]).…”

Section: Conflict Of Interestmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 88%

See 1 more Smart Citation

On Average Distance of Neighborhood Graphs and Its Applications

Mwakilama¹,

Ali²,

Chidzalo³

et al. 2022

Recent Applications in Graph Theory

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“…Das [9], in 2022, investigated various structural similarities between G and N (G) [3]. See [10][11][12] for more results related to neighborhood set.…”

Section: Introductionmentioning

confidence: 99%

The Transversal Neighborhood Domination Number on Parachute Graph and Semi-Parachute Graph

Fran,

Kusumastuti,

Elisa

et al. 2023

Advances in Physics Research

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Given V be a set of vertices on a graph G. A set D ⊆ V, is dominating set of G = (V, E) if all the vertex that is not in the set D are neighbors to at least one vertex of D. The smallest number of elements in D is known as the domination number. A set D ⊆ V(G) is called a transversal neighborhood-dominating set if D is the dominating set of G and intersects with every minimum neighborhood set. The transversal neighborhood domination number of G is the smallest number of elements in each transversal neighborhood-dominating set.. In this paper, we discuss transversal neighborhood domination number on a parachute graph and a semi-parachute graph. The transversal neighborhood domination number on parachute graph is (n + 3)/3 for every n = 3 k, (n + 5)/3 for every n = 3 k + 1, and (n + 4)/3 for every n = 3 k + 2. The neighborhood transversal domination number on the semi-parachute graph is (n + 1)/2 for every n = 2 k + 1 and (n + 2)/2 for every n = 2 k.

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On the complexity of the independent set problem in triangle graphs

Orlovich

Błażewicz²,

Dolgui³

et al. 2011

Discrete Mathematics

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International audienceWe consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G − I, there is a vertex w ∈ I such that {u, v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NPcomplete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P = NP

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Some Parameters on Neighborhood Number of A Graph

Cited by 4 publications

References 2 publications

On Average Distance of Neighborhood Graphs and Its Applications

On Average Distance of Neighborhood Graphs and Its Applications

The Transversal Neighborhood Domination Number on Parachute Graph and Semi-Parachute Graph

On the complexity of the independent set problem in triangle graphs

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