🤖 AI Summary
This study investigates the existence of two completely independent spanning trees (CISTs) in $k$-outerplanar triangulations. Through path-disjointness analysis, structural induction, and constructive proofs, it establishes for the first time that every 3-connected 2-outerplanar triangulation admits two CISTs. For the 3-outerplanar case, the work provides a sufficient condition guaranteeing the existence of such trees. Moreover, it constructs the first known counterexample—a 4-outerplanar triangulation that does not contain two CISTs. These results precisely delineate the boundary for CIST existence in low-level outerplanar graphs, significantly advancing the structural understanding of this important graph class.
📝 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.