Further results on the j-independence number of graphs

Document Type : Original paper

Authors

LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria

Abstract

In a graph $G$ of minimum degree $\delta$ and maximum degree $\Delta$, a subset $S$ of vertices of $G$ is $j$-independent, for some positive integer $j,$ if every vertex in $S$ has at most $j-1$ neighbors in $S$. The $j$-independence number $\beta_{j}(G)$ is the maximum cardinality of a $j$-independent set of $G$. We first establish an inequality between $\beta_{j}(G)$ and $\beta_{\Delta}(G)$ for $1\leq j\leq\delta-1$. Then we characterize all graphs $G$ with $\beta_{j}(G)=\beta_{\Delta}(G)$ for $j\in\{1,\dots,\Delta-1\}$, where the particular cases $j=1,2,\delta-1$ and
$\delta$ are well distinguished.

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References

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Communications in Combinatorics and Optimization
Volume 9, Issue 1
March 2024
Pages 1-11

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