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Partial Domination in Graphs
Combinatorics · Submitted 16 Jul 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, we introduce $\alpha$-partial domination number in a graph $G$ and study different bounds on the partial domination number of a graph $G$ with respect to its order, maximum degree, domination number etc., Moreover, $\alpha$-partial domination spectrum is introduced and Nordhaus-Gaddum bounds on the partial domination number are studied.
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The abstract, sentence by sentence
Let $G=(V,E)$ be a graph. For some $\alpha$ with $0<\alpha \leq 1$, a subset $S$ of $V$ is said to be a $\alpha$-partial dominating set if $|N[S]|\geq \alpha |V|$. The size of a smallest such $S$ is called the $\alpha$-partial domination number and is denoted by $\mathsf{pd}_\alpha(G)$. In this paper, we introduce $\alpha$-partial domination number in a graph $G$ and study different bounds on the partial domination number of a graph $G$ with respect to its order, maximum degree, domination number etc., Moreover, $\alpha$-partial domination spectrum is introduced and Nordhaus-Gaddum bounds on the partial domination number are studied.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $G=(V,E)$ be a graph. | background 1.00 | 0.04 | 0.01 | asserted 0.66 |
| s1 | For some $\alpha$ with $0<\alpha \leq 1$, a subset $S$ of $V$ is said to be a $\alpha$-partial dominating set if $|N[S]|\geq \alpha |V|$. | background 0.94 | 0.12 | 0.03 | asserted 0.99 |
| s2 | The size of a smallest such $S$ is called the $\alpha$-partial domination number and is denoted by $\mathsf{pd}_\alpha(G)$. | background 0.92 | 0.19 | 0.03 | asserted 0.99 |
| s3 | In this paper, we introduce $\alpha$-partial domination number in a graph $G$ and study different bounds on the partial domination number of a graph $G$ with respect to its order, maximum degree, domination number etc., Moreover, $\alpha$-partial domination spectrum is introduced and Nordhaus-Gaddum bounds on the partial domination number are studied. | contribution 1.00 | 0.94 | 0.38 | asserted 0.99 |
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Original abstract
Let $G=(V,E)$ be a graph. For some $\alpha$ with $0<\alpha \leq 1$, a subset $S$ of $V$ is said to be a $\alpha$-partial dominating set if $|N[S]|\geq \alpha |V|$. The size of a smallest such $S$ is called the $\alpha$-partial domination number and is denoted by $\mathsf{pd}_\alpha(G)$. In this paper, we introduce $\alpha$-partial domination number in a graph $G$ and study different bounds on the partial domination number of a graph $G$ with respect to its order, maximum degree, domination number etc., Moreover, $\alpha$-partial domination spectrum is introduced and Nordhaus-Gaddum bounds on the partial domination number are studied.
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How the literature uses it
2 arXiv papers cite this in 2 sentences
Papers citing it
Cites it as background 2 papers
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Around the same time, in an independent work of Das the same concept was introduced [ref] .
Properties of Partial Dominating Sets of Graphs · Introduction · 0.96
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A closely related problem is the partial MDS (p-MDS) problem, which has been proposed already in the literature [ref] .
Target observation of complex networks · Introduction · 0.88
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