Cycles through a set of specified vertices of a planar graph

Abstract

Confirming a conjecture of Plummer, Thomas and Yu proved that a 4-connected planar graph contains a cycle through all but two (freely choosable) vertices.Here we prove that a planar graph $G$ contains a cycle through $X\setminus \{x_1,x_2\}$ if $X\subseteq V(G)$, $X$ large enough, $x_1,x_2\in X$, and $X$ cannot be separated in $G$ by removing less than 4 vertices.