Year: 2026
Type: preprint
Abstract: We study the disjoint separators problem in graphs, an analogue of the famous disjoint paths problem. Given a graph $G$ and four pairwise disjoint subsets of vertices $S_r$, $T_r$, $S_b$, $T_b$, we ask whether there exist an $(S_r,T_r)$-separator and an $(S_b,T_b)$-separator which are disjoint. This is equivalent to coloring the vertices in red or blue, with $S_r \cup T_r$ in red and $S_b \cup T_b$ in blue, such that there is no red $(S_r,T_r)$-path and no blue $(S_b,T_b)$-path. On the one hand, we show that the disjoint separators problem is NP-complete. We actually exhibit several NP-complete restrictions of this problem, including planar graphs of bounded maximum degree, and graphs of bounded maximum degree when $|S_r|=|T_r|=|S_b|=|T_b|=1$. On the other hand, these hardness results turn out to be quite tight, as we provide a structural characterization and a polynomial-time algorithm for planar graphs when $|S_r|=|T_r|=|S_b|=|T_b|=1$. This has an interesting consequence about the popular board game Hex: for the generalized game that may be played on any board, our result characterizes the planar boards on which draws are impossible, thus extending the well-known result about impossibility of draws on the standard commercialized board.
Source: arXiv (Cornell University)
Institutions: Centre National de la Recherche Scientifique, Université de Montpellier, Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier, Aix-Marseille Université, Institut de Mathématiques de Marseille, Institut Polytechnique de Bordeaux, Lyon 1 Université, Laboratoire d'Informatique en Images et Systèmes d'Information, Laboratoire des Sciences du Numérique de Nantes, IMT Atlantique, Nantes Université, University of Clermont Auvergne, Clermont Université
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7167586928
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