The point-thicknesses of complete multipartite graphs

The point-thickness ${\theta }^{\mathrm{\prime }}(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ of $G$ is partitioned such that each subset induces a planar subgraph. In this paper, we determine the point-thickness of complete multipartite graphs. As a special case, we also obtain the point-thickness of complete graphs.