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arXiv · 2607.27802

Short Cycles Decide P-versus-NPC Status ofHamiltonicity on Bisplit Graphs

Mahendra Kumar R·Renjith P·Aadhavan S·Sadagopan N
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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.

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Mahendra Kumar R, Renjith P, Aadhavan S, Sadagopan N. 2026-07-30. Short Cycles Decide P-versus-NPC Status ofHamiltonicity on Bisplit Graphs. https://arxiv.org/abs/2607.27802

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