On the k-restricted Intersection Graph

Abstract

The problem on intersection graph was introduced by Szpilrajn-Marczewski in 1945. This study introduces a new variant of intersection graph, called the $k$-restricted intersection graph. Let $\mathcal{S}_n$ be a nonempty $n$-element set, for some positive integer $n$ and let $S_{(n,k)}$ be the set of all the $k$-element subsets of $\mathcal{S}_n$ where $0\leq k\leq n$. A $k$-restricted intersection graph, denoted by $G_{S_{(n,k)}}$, is a graph with vertex set  $S_{(n,k)}$ such that two vertices $A,B\in S_{(n,k)}$ are adjacent whenever $A\cap B\neq \emptyset$ and $A\neq B$. Here, we determined the order and size of $G_{S_{(n,k)}}$. Moreover, some parameters such as independence number, domination number, and isolate domination number of the $k$-restricted intersection graph were established. Finally, necessary and sufficient conditions for a $G_{S_{(n,k)}}$ to be isomorphic to the cycle graph and complete graph were determined.