Efficiently Enumerating All Spanning Trees of a Plane 3-Tree

Abstract

A spanning tree T of a connected, undirected graph G is an acyclic subgraph having all vertices and a minimal number of edges of G connecting those vertices. Enumeration of all possible spanning trees of undirected graphs is a well-studied problem. Solutions exist for enumeration for both weighted and unweighted graphs. However, these solutions are either space or time efficient. In this paper, we give an algorithm for enumerating all spanning trees in a plane 3-tree that is optimal in both time and space. Our algorithm exploits the structure of a plane 3-tree for a conceptually simpler alternative to existing general-purpose algorithms and takes $\mathcal {O}(n + m + \tau )$ time and $\mathcal {O}(n)$ space, where the given graph has n vertices, m edges, and $\tau$ spanning trees. This is a substantial improvement in both time and space complexity compared to the best-known algorithms for general graphs. We also propose a parallel algorithm for enumerating spanning trees of a plane 3-tree that has $\mathcal {O}(n + m + \frac{n\tau }{p})$ time and $\mathcal {O}(\frac{n\tau }{p})$ space complexities for p parallel processors. This second algorithm is useful when storing the spanning trees is important.