NP-Completeness of Hamiltonian Cycle Problem on Rooted Directed Path Graphs

TL;DR

This paper proves that determining the existence of a Hamiltonian cycle remains NP-Complete even in the specialized class of rooted directed path graphs, resolving an open problem in graph theory.

Contribution

It establishes NP-Completeness of the Hamiltonian Cycle Problem specifically for rooted directed path graphs, filling a gap in computational complexity knowledge.

Findings

NP-Completeness of Hamiltonian Cycle on rooted directed path graphs

Completes the classification of Hamiltonian cycle complexity in path graph subclasses

Provides a basis for understanding computational limits in specialized graph classes

Abstract

The Hamiltonian cycle problem is to decide whether a given graph has a Hamiltonian cycle. Bertossi and Bonuccelli (1986, Information Processing Letters, 23, 195-200) proved that the Hamiltonian Cycle Problem is NP-Complete even for undirected path graphs and left the Hamiltonian cycle problem open for directed path graphs. Narasimhan (1989, Information Processing Letters, 32, 167-170) proved that the Hamiltonian Cycle Problem is NP-Complete even for directed path graphs and left the Hamiltonian cycle problem open for rooted directed path graphs. In this paper we resolve this open problem by proving that the Hamiltonian Cycle Problem is also NP-Complete for rooted directed path graphs.