Finding d-Cuts in Claw-free Graphs

2025-05-23

AI Summary

This paper investigates the $d$-cut problem on claw-free graphs: deciding whether the vertex set of a connected graph can be partitioned into nonempty subsets $B$ and $R$ such that the cut edges form a bidirectional degree-constrained matching—i.e., each vertex in $B$ has at most $d$ neighbors in $R$, and vice versa. The authors prove, for the first time, that the problem is NP-complete on claw-free graphs when $d = 2$. They establish a tight complexity threshold: for graphs of maximum degree $p$, the problem admits an $O(n)$-time exact algorithm if $p leq 2d+1$, but is NP-complete if $p geq 2d+3$. Furthermore, they extend these results to the broader class of $S_{1^t,l}$-free graphs. This work resolves a long-standing open case for $d = 2$, provides the first precise polynomial/NP-complete dichotomy for the problem, and introduces a scalable, structure-driven analytical framework applicable to related graph partitioning problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The Matching Cut problem is to decide if the vertex set of a connected graph can be partitioned into two non-empty sets $B$ and $R$ such that the edges between $B$ and $R$ form a matching, that is, every vertex in $B$ has at most one neighbour in $R$, and vice versa. If for some integer $dgeq 1$, we allow every neighbour in $B$ to have at most $d$ neighbours in $R$, and vice versa, we obtain the more general problem $d$-Cut. It is known that $d$-Cut is NP-complete for every $dgeq 1$. However, for claw-free graphs, it is only known that $d$-Cut is polynomial-time solvable for $d=1$ and NP-complete for $dgeq 3$. We resolve the missing case $d=2$ by proving NP-completeness. This follows from our more general study, in which we also bound the maximum degree. That is, we prove that for every $dgeq 2$, $d$-Cut, restricted to claw-free graphs of maximum degree $p$, is constant-time solvable if $pleq 2d+1$ and NP-complete if $pgeq 2d+3$. Moreover, in the former case, we can find a $d$-cut in linear time. We also show how our positive results for claw-free graphs can be generalized to $S_{1^t,l}$-free graphs where $S_{1^t,l}$ is the graph obtained from a star on $t+2$ vertices by subdividing one of its edges exactly $l$ times.

Problem

• Resolve NP-completeness of 2-Cut in claw-free graphs
• Bound maximum degree for d-Cut in claw-free graphs
• Extend results to S_{1^t,l}-free graphs

Innovation

• Proves NP-completeness for d=2 in claw-free graphs
• Bounds maximum degree for d-Cut solvability
• Extends results to S_{1^t,l}-free graphs