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  "path": "/abs/2606.07453v1",
  "publishedAt": "2026-06-08T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Akramah Faizi",
    "Arash Rafiey"
  ],
  "textContent": "**Authors:** Akramah Faizi, Arash Rafiey\n\nThe Odd Cycle Transversal (OCT) problem, which asks for a minimum subset of vertices whose removal renders a graph bipartite, is a central problem in algorithmic graph theory. It is known to be NP-complete even on $P_k$-free graphs for $k \\ge 6$. Furthermore, assuming the Unique Games Conjecture (UGC), OCT does not admit a constant-factor approximation algorithm on general graphs. Motivated by these hardness results, we investigate the approximability of OCT on $P_k$-free graphs. We first establish that the problem becomes polynomial-time solvable on specific subclasses of $P_k$-free graphs, most notably $(P_6, C_3)$-free graphs, by exploiting a structural decomposition into rings of bipartite graphs. Leveraging these tractable substructures as a basis, we present a constant-factor approximation algorithm for OCT on general $P_k$-free graphs. We achieve an approximation ratio of $k-2$ when $k$ is odd and $k-3$ when $k$ is even. These results provide the first nontrivial constant-factor approximations for this class dependent on $k$, aligning with the UGC implication that no approximation factor independent of $k$ is likely to exist.",
  "title": "Odd Cycle Transversal in $P_k$-Free Graphs"
}