The influence of separating cycles in drawings of K ₅ ∖ e in the join product with paths and cycles

Abstract

The crossing number cr(H) of a graph H is the minimum number of edge crossings over all drawings of H in the plane. Let H ∗ be the connected graph of order five isomorphic to K 5 ∖ e obtained by removing one edge from the complete graph K 5. The main aim of the paper is to give the crossing numbers of the join products H ∗ + Pn and H ∗ + Cn , where Pn and Cn are the path and the cycle on n vertices, respectively. The proofs are done with the help of a suitable classification of a large number of drawings of the graph H ∗ in view of the existence of a separating cycle of two possible types.