On the genus of joins and compositions of graphs Original Research Article

We will use a surgical technique to imbed the composition graph G[H] when G has minimum degree 2 and H has even order. This imbedding is shown to be minimal when G is triangle-free. Along the way, we construct genus imbeddings of the join G + H when both factors have even order, G is empty or 1-regular, and G has order at least twice that of H. Variants and applications to specific graphs, such as complete tripartite graphs, are given.