Finding $d$-Cuts in Probe $H$-Free Graphs

2025-05-28

AI Summary

This paper studies the $d$-Cut problem in the probe graph model: given an integer $d geq 1$, determine whether a graph admits an edge cut such that each vertex has at most $d$ neighbors across the cut ($d=1$ corresponds to Matching Cut). Specifically, the input is a partitioned probe $H$-free graph $(G,P,N)$, where $N$ is an independent set and edges may be added within $N$ to make $G$ $H$-free. This is the first systematic extension of the $d$-Cut problem to the probe graph framework. Through combinatorial structural analysis, probe elimination techniques, and complexity classification methods, we fully characterize the computational complexity of the problem for all graphs $H$ and all $d geq 1$, precisely determining whether it is in P or NP-complete. Our results unify and strictly generalize all previously known complexity dichotomies for $d$-Cut on classical $H$-free graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

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
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

For an integer $dgeq 1$, the $d$-Cut problem is that of deciding whether a graph has an edge cut in which each vertex is adjacent to at most $d$ vertices on the opposite side of the cut. The $1$-Cut problem is the well-known Matching Cut problem. The $d$-Cut problem has been extensively studied for $H$-free graphs. We extend these results to the probe graph model, where we do not know all the edges of the input graph. For a graph $H$, a partitioned probe $H$-free graph $(G,P,N)$ consists of a graph $G=(V,E)$, together with a set $Psubseteq V$ of probes and an independent set $N=Vsetminus P$ of non-probes such that we can change $G$ into an $H$-free graph by adding zero or more edges between vertices in $N$. For every graph $H$ and every integer $dgeq 1$, we completely determine the complexity of $d$-Cut on partitioned probe $H$-free graphs.

Problem

• Determine complexity of d-Cut in probe H-free graphs
• Study edge cuts with vertex constraints in graphs
• Extend d-Cut results to partitioned probe H-free graphs

Innovation

• Probe graph model for H-free graphs
• Dynamic edge addition in non-probes
• Complexity analysis for d-Cut problem