The disjoint separators problem in graphs

2026-07-03

AI Summary

This study addresses the problem of determining whether two pairs of vertex sets in a graph admit disjoint separators, which is equivalent to the existence of a red-blue coloring such that no red path connects vertices within the red sets and no blue path connects vertices within the blue sets. The problem carries both theoretical significance and applications in game theory, notably in establishing draw-free conditions for generalized Hex boards. Through graph-theoretic modeling, complexity reductions, and structural decomposition, this work provides the first systematic characterization of its computational complexity: it proves NP-completeness for general graphs, planar graphs, and bounded-degree graphs. However, when the graph is planar and each set consists of a single vertex, the paper identifies a structural characterization and presents a polynomial-time decision algorithm.

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.

Problem

• disjoint separators
• graph theory
• NP-completeness
• planar graphs
• vertex separation

Innovation

• disjoint separators
• NP-completeness
• planar graphs
• polynomial-time algorithm
• Hex game