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

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 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 ≥ 4. Further, we also present a parallel algorithm for spanning tree enumeration using O(2) 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.