Domination and Total domination in Wrapped Butterfly networks

Shalini, V.; Rajasingh, Indra

doi:10.1016/j.procs.2020.05.010

Cited by 4 publications

(3 citation statements)

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“…Inverse domination number of circulant graph proved by V.Cynthiya in [3]. Also we identified domination and inverse domination numbers for Wrapped butterfly network, Lollipop graph, Fly graph and Jellyfish graph in [19][20][21].…”

Section: Introductionmentioning

confidence: 79%

Inverse Domination in X-Trees and Sibling Trees

Shalini,

Rajasingh

2024

Eur. J. Pure Appl. Math.

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A set $D$ of vertices in a graph $G$ is a dominating set if every vertex not in $D$ is adjacent to at least one vertex in $D$. The minimum cardinality of a dominating set in $G$ is called the domination number and is denoted by $\gamma(G)$. Let $D$ be a minimum dominating set of $G$. If $V-D$ contains a dominating set say $D^{'}$ of $G$, then $D^{'}$ is called an inverse dominating set with respect to $D$. The inverse domination number $\gamma^{'}(G)$ is the cardinality of a minimum inverse dominating set of $G$. A dominating set $D$ is called a connected dominating set or an independent dominating set of $G$ according as the induced subgraph $\langle D \rangle$ is connected or independent in $G$. The minimum of the cardinalities of the connected dominating sets of $G$ or the independent dominating sets of $G$ is called the connected domination number $\gamma_{c} (G)$ or the independent domination number $\gamma_{i} (G)$ respectively. In this paper, we determine the inverse domination numbers in X-Trees and Sibling Trees. We have also determined the independent domination numbers of both the trees and the connected domination number of Sibling Trees. A result on inverse domination number of some classes of Hypertrees is also included.

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Section: Introductionmentioning

confidence: 79%

Inverse Domination in X-Trees and Sibling Trees

Shalini,

Rajasingh

2024

Eur. J. Pure Appl. Math.

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“…Definition 3.1.1 (3) The vertex set of an n− dimensional butterfly network BF(k) is V = {(y; j)/ y = (y 1 , y 2 , . .…”

Section: Dom-chromatic Number Of Wrapped Butterfly Network W B(k)mentioning

confidence: 99%

“…The study on domination and coloring in wrapped butterfly and bloom graphs are interesting areas of research in graph theory. In wrapped butterfly network, few studies like 1-harmonious coloring (1) , T-coloring (2) , ST-coloring (2) , total domination (3) and domination of its digraph (4) were carried on. In the above research work, only one concept, either coloring or domination were applied, whereas in this paper both the concepts are applied together to determine the dom-chromatic number of wrapped butterfly graph.…”

Section: Introductionmentioning

confidence: 99%

Dom-Chromatic Number o f Wrapped Butterfly & Bloom Graphs

Punitha,

Angeline

2023

IJST

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Objectives: For a given graph G with proper coloring, the problem of selecting a dom-coloring set is to choose a dominating set having a property that it has a minimum of one vertex from every possible color class in G. Our aim is to determine the family of networks that allow dom-coloring and to find its dom-chromatic number denoted by γ dc (G). Method: We have applied the algorithmic method of choosing the dom-coloring set(dc-set). Here we have designed a coloring algorithm to yield the proper coloring for the vertices of the graph. D-set algorithm has been developed to determine the dominating set for the given graph. Then, the dc-set for the graph is obtained by applying the above two algorithms. Findings: In this study, we have established the study on finding the dc-set of wrapped butterfly network and bloom graphs. Further, we have found the dom-chromatic number of the above-mentioned graphs. Novelty: Dom-coloring is an extended variation of graph coloring and domination which has emerged as a result of the combination of the two broad concepts in graph theory namely, domination and coloring. A dominating set which includes a minimum of one vertex from all possible color classes of the graph forms a dom-coloring set. In this paper, a study on dom-coloring of wrapped butterfly and bloom graphs have been accomplished. These results may be generalized for butterfly derived networks to determine its domchromatic number.

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