Hamiltonian Path in $$K_{1,t}$$-free Split Graphs- A Dichotomy

Abstract

In this paper we investigate Hamiltonian path problem in the context of split graphs, and produce a dichotomy result on the complexity of the problem. Our main result is a deep investigation of the structure of K1,4-free split graphs in the context of Hamiltonian path problem, and as a consequence, we obtain a polynomial-time algorithm to the Hamiltonian path problem in K1,4-free split graphs. We close this paper with the hardness result: we show that, unless P=NP, Hamiltonian path problem is NP-complete in K1,5-free split graphs by reducing from Hamiltonian cycle problem in K1,5-free split graphs. Thus this paper establishes a "thin complexity line" separating NP-complete instances and polynomial-time solvable instances.