On the k-restricted Intersection Graph

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 Sₙ 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 Sₙ where 0≤ k≤ 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∈ S(n,k) are adjacent whenever A∩ B≠ ∅ and A≠ 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.