Joins of 1-Planar Graphs

A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study 1-planar graph joins. We prove that the join G + H is 1-planar if and only if the pair [G, H] is subgraph-majorized by one of pairs [C-3 boolean OR C-3, C-3], [C-4, C-4], [C-4, C-3], [K-2,K-1,K-1, P-3] in the case when both elements of the graph join have at least three vertices. If one element has at most two vertices, then we give several necessary/sufficient conditions for the bigger element.