Listing All Spanning Trees in Halin Graphs - Sequential and Parallel view

TL;DR

This paper introduces sequential and parallel algorithms for listing all spanning trees in Halin graphs, efficiently handling the enumeration without repetitions and analyzing the computational complexity.

Contribution

It presents the first known algorithms for enumerating all spanning trees in Halin graphs, including a parallel approach with processor complexity analysis.

Findings

Enumeration without repetitions achieved

Parallel algorithm uses $O((2pd)^p)$ processors

Number of spanning trees in Halin graphs is $O((2pd)^p)$

Abstract

For a connected labelled graph $G$, a {\em spanning tree} $T$ is a connected and an acyclic subgraph that spans all vertices of $G$. In this paper, we consider a classical combinatorial problem which is to list all spanning trees of $G$. A Halin graph is a graph obtained from a tree with no degree two vertices and by joining all leaves with a cycle. We present a sequential and parallel algorithm to enumerate all spanning trees in Halin graphs. Our approach enumerates without repetitions and we make use of $O((2pd)^{p})$ processors for parallel algorithmics, where $d$ and $p$ are the depth, the number of leaves, respectively, of the Halin graph. We also prove that the number of spanning trees in Halin graphs is $O((2pd)^{p})$.