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2022•Discrete Mathematics Algorithms and ApplicationsRequires access

2-Trees: Structural insights and the study of Hamiltonian paths

P. Renjith, N. Sadagopan

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Abstract

For a connected graph, a path containing all vertices is known as a 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.

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What this paper is about

For a connected graph, a path containing all vertices is known as a 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.

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Available abstract

For a connected graph, a path containing all vertices is known as a 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.

Key concepts: Hamiltonian path, Hamiltonian path problem, Hamiltonian (control theory), Mathematics, Combinatorics, Trémaux tree, Graph, Path (computing)

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