Listing spanning trees of outerplanar graphs by pivot-exchanges

TL;DR

The paper addresses listing spanning trees of outerplanar graphs using pivot-exchanges and related face-exchanges, connecting Gray-code generation to the 0/1-polytope skeleton of spanning-tree incidence vectors. It introduces Algorithm G with a dual-based edge labeling that yields genlex Gray codes: a genlex pivot-exchange code for outerplane triangulations and a genlex pivot$\vee$-face code for outerplane graphs, with $O(n\log n)$ time per tree and $O(n)$ space. It also proves a Fibonacci upper bound $t(G)\le f_{m+1}$ for the number of spanning trees in outerplane (multi)graphs, with equality characterizations for triangulations whose weak dual is a path. The results fuse combinatorial labeling techniques with duality to produce efficient, history-free Gray-code listings and illuminate extremal structures, while leaving open questions about broader graph classes and directed analogues.

Abstract

We prove that the spanning trees of any outerplanar triangulation $G$ can be listed so that any two consecutive spanning trees differ in an exchange of two edges that share an end vertex. For outerplanar graphs $G$ with faces of arbitrary lengths (not necessarily 3) we establish a similar result, with the condition that the two exchanged edges share an end vertex or lie on a common face. These listings of spanning trees are obtained from a simple greedy algorithm that can be implemented efficiently, i.e., in time $\mathcal{O}(n \log n)$ per generated spanning tree, where $n$ is the number of vertices of $G$. Furthermore, the listings correspond to Hamilton paths on the 0/1-polytope that is obtained as the convex hull of the characteristic vectors of all spanning trees of $G$.

Figures (14)

• Figure 1: The spanning tree flip graph ${\mathcal{F}}(G)$ for the 'diamond' graph $G$ on the left, with a Hamilton cycle highlighted. For each spanning tree, the set of edges is shown above, and the characteristic vector is shown below. Edges of ${\mathcal{F}}(G)$ are labelled by the two edges of $G$ being exchanged.
• Figure 2: Three different edge exchange Gray codes for listing the 21 spanning trees of the fan graph $F_5$. In each spanning tree, the edge removed to reach the next tree is highlighted (prefixed by $-$ in (c)), and the non-edge being added is dashed (prefixed by $+$ in (c)). In (b) and (c), the common end vertex of each pivot-exchange operation is highlighted, and in (c), the common face of each face-exchange operation is highlighted. The right-hand side of (c) shows the characteristic vectors of each spanning tree.
• Figure 3: The fan graphs $F_n$.
• Figure 4: Illustration of the sixth iteration of Algorithm G in the run shown in Figure \ref{['fig:3GCs']} (c).
• Figure 5: Connection between pivot- and face-exchanges via the dual spanning tree.

Theorems & Definitions (20)

• Theorem 3: MR4756593
• Theorem 4: MR4473269
• Theorem 6
• Theorem 7
• Theorem 8
• Corollary 9
• Theorem 10
• Remark 11
• Remark 12
• Lemma 13