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DOMINATION DEFECT IN THE COMPOSITION OF GRAPHS
Keywords:
k-domination defect, minimum dominating set, composition of graphs.DOI:
https://doi.org/10.17654/0974165823048Abstract
Let $G=(V(G), E(G))$ be a simple connected graph of order $n$ with domination number $\gamma(G) \geq 2$ and let $1 \leq k<\gamma(G)$. A $k$-domination defect set of $G$ is a nonempty set $S \subseteq V(G)$ of cardinality $\gamma(G)-k$ such that $\left|V(G) \backslash N_G[S]\right|$ is minimum among the subsets of $V(G)$ containing $\gamma(G)-k$ vertices. The minimum number of vertices in $G$ which are left undominated by $S$ is denoted by $\zeta_k(G)$. This paper presents results on the domination defect of graphs resulting from the composition product $G[H]$ of a connected graph $G$ and any graph $H$. In particular, the $k$-domination defect set of $G[H]$ is characterized and $\zeta_k(G[H])$ is determined.
Received: April 28, 2023
Revised: May 20, 2023
Accepted: June 8, 2023
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