On a graph isomorphic to its intersection graph: self-graphoidal graphs

A graph $G$ is called a graphoidal graph if there exists a graph $H$ and a graphoidal cover $\psi$ of $H$ such that $G\cong\Omega(H,\psi)$. Then the graph $G$ is said to be self-graphoidal if it is isomorphic to one of its graphoidal graphs. In this paper, we have examined the existence of a few self-graphoidal graphs from path length sequence of a graphoidal cover and obtained new results on self-graphoidal graphs.