Connectivity-Preserving Minimum Separator in AT-free Graphs
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916780285362176 |
|---|---|
| author | Kenig, Batya |
| author_facet | Kenig, Batya |
| contents | Let $A$ and $B$ be disjoint, non-adjacent vertex-sets in an undirected, connected graph $G$, whose vertices are associated with positive weights. We address the problem of identifying a minimum-weight subset of vertices $S\subseteq V(G)$ that, when removed, disconnects $A$ from $B$ while preserving the internal connectivity of both $A$ and $B$. We call such a subset of vertices a connectivity-preserving, or safe minimum $A,B$-separator. Deciding whether a safe $A,B$-separator exists is NP-hard by reduction from the 2-disjoint connected subgraphs problem, and remains NP-hard even for restricted graph classes that include planar graphs, and $P_\ell$-free graphs if $\ell\geq 5$. In this work, we show that if $G$ is AT-free then in polynomial time we can find a safe $A,B$-separator of minimum weight, or establish that no safe $A,B$-separator exists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Connectivity-Preserving Minimum Separator in AT-free Graphs Kenig, Batya Data Structures and Algorithms Let $A$ and $B$ be disjoint, non-adjacent vertex-sets in an undirected, connected graph $G$, whose vertices are associated with positive weights. We address the problem of identifying a minimum-weight subset of vertices $S\subseteq V(G)$ that, when removed, disconnects $A$ from $B$ while preserving the internal connectivity of both $A$ and $B$. We call such a subset of vertices a connectivity-preserving, or safe minimum $A,B$-separator. Deciding whether a safe $A,B$-separator exists is NP-hard by reduction from the 2-disjoint connected subgraphs problem, and remains NP-hard even for restricted graph classes that include planar graphs, and $P_\ell$-free graphs if $\ell\geq 5$. In this work, we show that if $G$ is AT-free then in polynomial time we can find a safe $A,B$-separator of minimum weight, or establish that no safe $A,B$-separator exists. |
| title | Connectivity-Preserving Minimum Separator in AT-free Graphs |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2506.03612 |