Linkage problem on optimal $1$-planar graphs

Abstract

Enami and Maezawa give a complete characterization of $(s_1,s_2,\dots ,s_k)$-linked planar graphs for any $k$-tuple of positive integers. In this paper, we investigate linkage problems for optimal 1-planar graphs. In particular, we show that every optimal 1-planar graph with connectivity $6$ is $(5,5)$-linked. Moreover, for an optimal $1$-planar graph $G$ that is not $(2,2,1)$-linked, we characterize disjoint vertex subsets $S_1,S_2,S_3$ in $G$ with $|S_1|=|S_2|=2$ and $|S_3|=1$ such that $G$ is not $\{S_1,S_2,S_3\}$-linked.