From the Crossing Numbers of K 5 + P n and K 5 + C n to the Crossing Numbers of W m + S n and W m + W n.

The crossing number of a graph is a significant measure that indicates the complexity of the graph and the difficulty of visualizing it. In this paper, we examine the crossing numbers of join products involving the complete graph K 5 with discrete graphs, paths, and cycles. We analyze optimal drawings of K 5 , identify all five non-isomorphic drawings, and address previously hypothesized crossing numbers for K 5 + P n , and K 5 + C n . Through a simplified approach, we first establish cr (K 5 + D n) and then extend our method to prove the crossing numbers cr (K 5 + P n) and cr (K 5 + C n) . These results also lead to new hypotheses for cr (W m + S n) and cr (W m + W n) involving wheels and stars. Our findings correct previous inaccuracies in the literature and provide a foundation for future research. [ABSTRACT FROM AUTHOR]