Completely Independent Spanning Trees in $k$-Outerplanar Triangulated Discs

Abstract

Let $T_1,T_2,\dots ,T_k$ be $k$ spanning trees of a graph $G$. For any pair of vertices $u$ and $v$, if the $u$--$v$ paths in the $k$ spanning trees are pairwise openly disjoint, then the spanning trees are called completely independent spanning trees (CISTs) of $G$. In this paper, we first prove that every 3-connected 2-outerplanar triangulated disc has two completely independent spanning trees. Next, for a 3-connected 3-outerplanar triangulated disc $G$, we provide sufficient conditions for $G$ to have two completely independent spanning trees. We provide an example of a 3-connected 4-outerplanar triangulation that does not have two completely independent spanning trees.