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ЖУРНАЛЫ // Математические заметки // Архив

Матем. заметки, 2022, том 111, выпуск 2, страницы 297–304 (Mi mzm13227)

Эта публикация цитируется в 1 статье

Статьи, опубликованные в английской версии журнала

Perfect Domination Polynomial of Homogeneous Caterpillar Graphs and of Full Binary Trees

Temesgen Engida Yimer, J. Baskar Babujee

Department of Mathematics, Anna University, MIT Campus, Chennai, 600044 India

Аннотация: Let $G=(V,E)$ be a simple graph of order $n$. A set $S \subseteq V(G)$ is a perfect dominating set of a graph $G$ if every vertex $v\in V(G)-S$ is adjacent to exactly one vertex in $S$. That is, every vertex outside $S$ has exactly one neighbor in $S$. Every graph $G$ has at least the trivial perfect dominating sets consisting of all vertices in $G$. The perfect domination number $\gamma_{pf} (G)$ is the minimal cardinality of dominating sets in $G$. Let $D_{pf} (G,i)$ be the family of perfect dominating sets for a graph $G$ with cardinality $i$ and $d_{pf} (G,i)= |D_{pf} (G,i)|$. The perfect domination polynomial of a graph $G$ of order $n$ is
$$ D_{pf} (G,x)=\sum_{i=\gamma_{pf}(G)}^{n} d_{pf}(G,i)x^n, $$
where $d_{pf} (G,i)$ is the number of perfect dominating sets of $G$ of size $i$. In this paper, we studied the perfect domination polynomial $D_{pf} (G,x)$ of homogeneous caterpillar graphs and of full binary trees.

Ключевые слова: perfect domination sets, perfect domination polynomial, homogeneous caterpillar graphs, full binary tree, corona graph.

Поступило: 16.07.2021

Язык публикации: английский


 Англоязычная версия: Mathematical Notes, 2022, 111:2, 297–304

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