The Hamiltonian Cycle in $K_{1,r}$-free Split Graphs -- A Dichotomy

TL;DR

This paper establishes a clear complexity dichotomy for the Hamiltonian cycle problem in split graphs, showing NP-completeness in certain classes and polynomial-time solvability in others, advancing understanding of graph structure impacts.

Contribution

The paper proves NP-completeness of HCYCLE in $K_{1,5}$-free split graphs and provides polynomial algorithms for $K_{1,3}$-free and $K_{1,4}$-free split graphs, revealing structural complexity boundaries.

Findings

NP-complete in $K_{1,5}$-free split graphs

Polynomial-time algorithms for $K_{1,3}$-free and $K_{1,4}$-free split graphs

Structural results suggest similar dichotomies for related problems

Abstract

In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete in planar bipartite graphs with maximum degree $3$. Using this reduction, we show that HCYCLE is NP-complete in split graphs. In particular, we show that the problem is NP-complete in $K_{1,5}$-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in $K_{1,3}$-free and $K_{1,4}$-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem (HPATH) and other variants of HCYCLE.