Fair domination in the join, corona and composition of graphs

Maravilla, Emiliano C.; Isla, Rowena T.; Canoy, Sergio R.

doi:10.12988/ams.2014.46397

Cited by 2 publications

(2 citation statements)

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“…Das and Desormeaux [6] considered the problem of minimizing the size of a regular dominating set that induces a connected subgraph. Further results on fair domination are obtained in [8,5].…”

Section: Introductionmentioning

confidence: 87%

Irregular independence and irregular domination

Borg

Caro

Fenech

2019

Discrete Applied Mathematics

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If A is an independent set of a graph G such that the vertices in A have different degrees, then we call A an irregular independent set of G. If D is a dominating set of G such that the vertices that are not in D have different numbers of neighbours in D, then we call D an irregular dominating set of G. The size of a largest irregular independent set of G and the size of a smallest irregular dominating set of G are denoted by α ir (G) and γ ir (G), respectively. We initiate the investigation of these two graph parameters. For each of them, we obtain sharp bounds in terms of basic graph parameters such as the order, the size, the minimum degree and the maximum degree, and we obtain Nordhaus-Gaddum-type bounds. We also establish sharp bounds relating the two parameters. Furthermore, we characterize the graphs G with α ir (G) = 1, we determine those that are planar, and we determine those that are outerplanar.

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Section: Introductionmentioning

confidence: 87%

Irregular independence and irregular domination

Borg

Caro

Fenech

2019

Discrete Applied Mathematics

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“…The Corona of two graphs G and H, represented as G • H, is created by taking a single copy of G and replicating H, |V(G)| times, with the i-th vertex of G connected to each vertex in the i-th replica of H. Previous studies of FD set in this binary operation can be seen in [9,10].…”

Section: Introductionmentioning

confidence: 99%

Fair Detour Domination in the Corona of Two Graphs: Optimizing CCTV Camera Installation for Efficient Surveillance

Ebenezer,

Parthipan

2024

J. Sci. Res.

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The objective of our research is to analyze the properties of FDD sets in the corona of two graphs and explore their practical application in CCTV camera installation. A set F ⊆ V(G) is considered to be a Fair Detour Dominating set (FDD-set) if it is detour dominating and the number of neighbors within set F is the same for any pair of vertices outside of F. Among these FDD sets, the fγd-set refers to the FDD-set with the smallest number of vertices, and its order defines the Fair Detour Domination number (fγd (G)). We have established that for any arbitrary graphs G1 and G2, fγd (G1 o G2) = |V(G1) iff fd(G2) = 1 we have determined the fγd number of corona products of any connected graph G with the path graph as well as the cycle graph. We also characterized FDD sets in the corona product of two connected graphs and provided a thorough description of how FDD sets can be used in optimizing CCTV camera installation for efficient surveillance.

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Fair domination in the join, corona and composition of graphs

Abstract: In this paper, we characterize the fair dominating sets in the join, corona and composition of graphs. We also determine the bounds or exact values of the fair domination numbers of these graphs.

Cited by 2 publications

References 4 publications

Irregular independence and irregular domination

Irregular independence and irregular domination

Fair Detour Domination in the Corona of Two Graphs: Optimizing CCTV Camera Installation for Efficient Surveillance

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