Short Cycles Decide P-versus-NPC Status ofHamiltonicity on Bisplit Graphs
arXiv:2607.27802
The paper classifies the complexity of Hamiltonian cycle and path problems on bisplit graphs, showing polynomial-time solvability for chordal bisplit graphs and NP-completeness for chordal bipartite bisplit graphs, with further results for P5‑free and P10‑free cases.
Abstract
A connected graph G is said to be a bisplit graph if the vertex set of G can be partitioned into a stable set and a complete bipartite graph. We establish the following dichotomy with chordality being the parameter; for chordal bisplit graphs, Hamiltonian cycle (HCYCLE) and Hamiltonian path (HPATH) problems are polynomial-time solvable, and for chordal bipartite bisplit graphs, HCYCLE (HPATH) is NP-complete. We further strengthen the result of [1] and show that HCYCLE (HPATH) is polynomial-time solvable on P5-free chordal bipartite graphs (bipartite chain graphs) and NP-complete on P10-free chordal bipartite graphs. By using our polynomial results on HCYCLE (HPATH) as a framework, we solve many variants and generalizations of HCYCLE (HPATH), which are also reported in this paper.