2026/07/03 by Thomas Delépine, Florian Galliot, Yannick Mogge +2 · 1 voice
Computer Science · Mathematics · #cs.DM #math.CO
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 Sr, Tr, Sb, Tb, we ask whether there exist an (Sr,Tr)-separator and an (Sb,Tb)-separator which are disjoint. This is equivalent to coloring the vertices in red or blue, with Sr ∪ Tr in red and Sb ∪ Tb in blue, such that there is no red (Sr,Tr)-path and no blue (Sb,Tb)-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 |Sr|=|Tr|=|Sb|=|Tb|=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 |Sr|=|Tr|=|Sb|=|Tb|=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.