Spanning Tree Enumeration in 2-trees: Sequential and Parallel Perspective

Abstract

For a connected graph, a vertex separator is a set of vertices whose removal creates at least two components. A vertex separator $S$ is minimal if it contains no other separator as a strict subset and a minimum vertex separator is a minimal vertex separator of least cardinality. A {\em clique} is a set of mutually adjacent vertices. A 2-tree is a connected graph in which every maximal clique is of size three and every minimal vertex separator is of size two. A spanning tree of a graph $G$ is a connected and an acyclic subgraph of $G$. In this paper, we focus our attention on two enumeration problems, both from sequential and parallel perspective. In particular, we consider listing all possible spanning trees of a 2-tree and listing all perfect elimination orderings of a chordal graph. As far as enumeration of spanning trees is concerned, our approach is incremental in nature and towards this end, we work with the construction order of the 2-tree, i.e. enumeration of $n$-vertex trees are from $n-1$ vertex trees, $n \geq 4$. Further, we also present a parallel algorithm for spanning tree enumeration using $O(2^n)$ processors. To our knowledge, this paper makes the first attempt in designing a parallel algorithm for this problem. We conclude this paper by presenting a sequential and parallel algorithm for enumerating all Perfect Elimination Orderings of a chordal graph.