🤖 AI Summary
This study delineates the computational complexity boundary of the Hamiltonian cycle and path problems on split–split graphs. By introducing chordality and forbidden induced path length (Pₖ-free) as key structural parameters, and leveraging graph decomposition, structural graph theory, and complexity reductions, the work precisely identifies the threshold at which the problem transitions from polynomial-time solvability to NP-completeness. The main contributions include proving that the problem is polynomial-time solvable on chordal split–split graphs, yet NP-complete even on chordal bipartite split–split graphs. Furthermore, the paper establishes a tight complexity dichotomy by showing tractability for P₅-free instances and intractability for P₁₀-free instances, extending these results to several related variants.
📝 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.