Hamiltonian Path in Split Graphs- a Dichotomy

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 $K_{1,4}$-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 $K_{1,4}$-free split graphs. We close this paper with the hardness result: we show that, unless P=NP, Hamiltonian path problem is NP-complete in $K_{1,5}$-free split graphs by reducing from Hamiltonian cycle problem in $K_{1,5}$-free split graphs. Thus this paper establishes a"thin complexity line"separating NP-complete instances and polynomial-time solvable instances.