On self-graphoidal graphs and their complements

The graphoidal graph G of graph H is the graph obtained by taking graphoidal cover Psi of H as vertices and two vertices are adjacent if and only if the corresponding paths have a non-empty intersection. If G is isomorphic to one of its graphoidal graphs, then G is said to be a self-graphoidal graph G is called self-complementary graphoidal graph if it is isomorphic to one of its complementary graphoidal graphs. In this article, we characterize self-graphoidal graphs and give a construction of self-graphoidal graphs from cycle and wheel graphs. Also, we give a characterization of self-complementary graphoidal graphs.