J-Open Independent Sets in Graphs (European Journal of Pure and Applied Mathematics, 2024)
Let $G eq \overline{K}_n$ be a graph with vertex and edge-sets $V(G)$ and $E(G)$, respectively. Then\linebreak $O$ $\subseteq$ $V(G)$ is called a J-open independent set of $G$ if for every $a,b \in V(G)$ where $a eq b$, $d_G(a,b)$ $ eq 1$, and $N_G(a) \backslash N_G(b) eq \varnothing$ and $N_G(b)\backslash N_G(a) eq \varnothing$. The maximum cardinality of a J-open independent set of G, denoted by $α_J(G)$, is called the J-open independence number of $G$. In this paper, we introduce new independence parameter called J-open independence. We show that this parameter is always less than or equal to the standard independence (resp. J-total domination) parameter of a graph. In fact, their differences can be made arbitrarily large. In addition, we show that J-open independence parameter is incomparable with hop independence parameter. Moreover, we derive some formulas and bounds of the parameter for some classes of graphs and the join of two graphs.
Keywords: J-Open Independent set; J-Open Independence number