Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 9 1 2024 03 01 Further results on the j-independence number of graphs 1 11 14479 10.22049/cco.2022.28012.1417 EN Ahmed Bouchou LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria Mustapha Chellali LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria Journal Article 2022 09 20 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<br />$\delta$ are well distinguished. https://comb-opt.azaruniv.ac.ir/article_14479_e04e66a5d2826a7c7f6179a4c1fc6ee6.pdf