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 (₍, ₊) be the set of all the k-element subsets of Sₙ where 0 k n. A k-restricted intersection graph, denoted by Gₒ_ (₍, ₊), is a graph with vertex set S (₍, ₊) such that two vertices A, B S (₍, ₊) are adjacent whenever A B and A B. Here, we determined the order and size of Gₒ_ (₍, ₊). 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 Gₒ_ (₍, ₊) to be isomorphic to the cycle graph and complete graph were determined.