2-Trees: Structural Insights and the study of Hamiltonian Paths

TL;DR

This paper provides a complete structural characterization and a linear-time algorithm for determining the existence of Hamiltonian paths specifically in 2-trees, a special class of graphs.

Contribution

It introduces a necessary and sufficient condition for Hamiltonian paths in 2-trees and presents a linear-time algorithm based on this characterization.

Findings

A structural characterization of Hamiltonian paths in 2-trees.

A linear-time algorithm for detecting Hamiltonian paths in 2-trees.

Potential applications of the combinatorial insights to other problems on 2-trees.

Abstract

For a connected graph, a path containing all vertices is known as \emph{Hamiltonian path}. For general graphs, there is no known necessary and sufficient condition for the existence of Hamiltonian paths and the complexity of finding a Hamiltonian path in general graphs is NP-Complete. We present a necessary and sufficient condition for the existence of Hamiltonian paths in 2-trees. Using our characterization, we also present a linear-time algorithm for the existence of Hamiltonian paths in 2-trees. Our characterization is based on a deep understanding of the structure of 2-trees and the combinatorics presented here may be used in other combinatorial problems restricted to 2-trees.