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For a connected graph, a path containing all vertices is known as\emph{Hamiltonian path}. For general graphs, there is no known necessary andsufficient condition for the existence of Hamiltonian paths and the complexityof finding a Hamiltonian path in general graphs is NP-Complete. We present anecessary and sufficient condition for the existence of Hamiltonian paths in2-trees. Using our characterization, we also present a linear-time algorithmfor the existence of Hamiltonian paths in 2-trees. Our characterization isbased on a deep understanding of the structure of 2-trees and the combinatoricspresented here may be used in other combinatorial problems restricted to2-trees.

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