Hamiltonian path in permutation graphs

Abstract

Hamiltonian path problem is NP-Complete for general graphs. There are some polynomial time algorithms for the problem in some special graphs, such as interval graphs, co-comparability graphs, etc. The existing best algorithm for the Hamiltonian path problem in a permutation graph requires $O\left(n^2\right)$ time. We introduce the notion of a follow-up vertex in a permutation graph with respect to its Hasse diagram. We also provide a necessary and sufficient condition for the existence of a Hamiltonian path in a permutation graph in terms of the existence of a follow-up vertex. This finding leads to the design of an improved algorithm for the Hamiltonian path problem in a permutation graph that takes $O(n\log \log n)$ time.