The $k$-independent graph of a graph

TL;DR

This paper studies the properties of the $k$-independent graph of a graph, which encodes relationships between independent sets of size at most $k$, and computes it for specific graph classes.

Contribution

It introduces and analyzes the $k$-independent graph, providing properties and explicit computations for certain graphs, expanding understanding of independent set structures.

Findings

Properties of $I_k(G)$ are established.

Explicit computations of $I_k(G)$ for some graphs are provided.

Insights into the structure of independent sets in relation to $k$.

Abstract

Let $G=(V,E)$ be a simple graph. A set $I\subseteq V$ is an independent set, if no two of its members are adjacent in $G$. The $k$-independent graph of $G$, $I_k (G)$, is defined to be the graph whose vertices correspond to the independent sets of $G$ that have cardinality at most $k$. Two vertices in $I_k(G)$ are adjacent if and only if the corresponding independent sets of $G$ differ by either adding or deleting a single vertex. In this paper, we obtain some properties of $I_k(G)$ and compute it for some graphs.