Year: 2026
Type: preprint
Abstract: A $u\hbox{-}v$ separator in $G$ is a set $S\subseteq V(G)\setminus\{u,v\}$ such that there is no $u\hbox{-}v$ path in $G\setminus S$. A minimal separator in $G$ is a set $S$ such that for some $u,v\in V(G)\setminus S$, $S$ is a $u\hbox{-}v$ separator in $G$ but no proper subset of $S$ is. A set $S$ is bisimplicial if it is the union of at most two cliques. Recognizing graphs in which every minimal separator is the union of at most $k$ cliques is polynomial-time solvable for $k=1$ and NP-hard for every fixed $k\geq 3$. We settle the remaining case $k=2$ by giving a polynomial-time algorithm to decide whether every minimal separator in a graph is bisimplicial. A set $S$ is triad-free if its independence number $α(S)$ is at most two. Given a graph $G$ and two vertices $u,v$, the problem of finding a $u\hbox{-}v$ separator $S$ with $α(S)\leq k$ had the same complexity gap: it was known to be polynomial-time solvable for $k=1$ and NP-hard for every fixed $k\geq 3$. We resolve the remaining case $k=2$ by giving a polynomial-time algorithm that finds a triad-free $u\hbox{-}v$ separator or determines that none exists.
Source: arXiv (Cornell University)
Authors: Chudnovsky, Maria, Maier, Ilya
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222364792