Efficient Algorithms for Finding d-Cuts in Graphs with Bounded Diameter, Bounded Radius, and H-Free Graphs

Core Concepts

The d-Cut problem, a generalization of the Matching Cut problem, can be solved efficiently for graphs of bounded diameter, bounded radius, and H-free graphs, with complexity differences compared to the d=1 case.

Abstract

The paper presents a systematic study of the complexity of the d-Cut problem, which generalizes the well-studied Matching Cut problem, for various special graph classes.
The key results are:

For d ≥ 2, d-Cut is polynomial-time solvable for graphs of diameter at most 2, (P3 + P4)-free graphs, and P5-free graphs. These results extend the known polynomial-time solvability of Matching Cut (d=1) for these graph classes.

However, the paper also shows several NP-hardness results for d-Cut that contrast with the known polynomial-time results for d=1. This includes NP-completeness for d ≥ 2 on graphs of bounded radius, 3P2-free graphs, and (H1, H2, ..., H*i)-free graphs for every i ≥ 1.

The results lead to full dichotomies for bounded diameter and bounded radius graphs, and almost-complete dichotomies for H-free graphs, showing complexity differences between the d=1 and d≥2 cases.

The paper develops novel algorithmic techniques and analysis to obtain these results, going beyond the approaches used for the d=1 case.

Stats

The graph Hi is obtained from the "H"-graph H1 by subdividing the edge uv exactly i-1 times.
The diameter of a graph G is the maximum eccentricity over all vertices of G.
The radius of a graph G is the minimum eccentricity over all vertices of G.

Quotes

"For d ≥ 2, d-Cut is polynomial-time solvable for graphs of diameter at most 2, (P3 + P4)-free graphs, and P5-free graphs."
"The paper also shows several NP-hardness results for d-Cut that contrast with the known polynomial-time results for d=1."

Key Insights Distilled From

Finding $d$-Cuts in Graphs of Bounded Diameter, Graphs of Bounded Radius and $H$-Free Graphs

by Feli... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2404.11389.pdf

Deeper Inquiries

How can the techniques developed in this paper be extended to obtain a complete complexity classification of the d-Cut problem for H-free graphs

To obtain a complete complexity classification of the d-Cut problem for H-free graphs, we can build upon the techniques developed in the paper and extend them to cover all possible cases for different graph classes. This would involve systematically studying the complexity of the d-Cut problem for various classes of H-free graphs by considering different forbidden induced subgraphs and structural properties. By analyzing the behavior of the d-Cut problem on a wider range of H-free graph classes, we can identify patterns and characteristics that determine the computational complexity of the problem. This comprehensive study would involve proving polynomial-time solvability and NP-completeness results for different graph classes within the H-free category, leading to a complete dichotomy for the d-Cut problem in H-free graphs.

What other structural graph parameters, besides diameter and radius, could lead to interesting complexity differences between the d=1 and d≥2 cases of the d-Cut problem

Besides diameter and radius, several other structural graph parameters could lead to interesting complexity differences between the cases of d=1 and d≥2 in the d-Cut problem. Some of these parameters include:

Degree of vertices: Analyzing the impact of vertex degrees on the complexity of the d-Cut problem could reveal interesting insights. For instance, studying the problem for graphs with bounded maximum degree or average degree could lead to distinct complexity results.
Connectivity: Investigating the connectivity properties of graphs, such as cut vertices, bridges, or biconnected components, could influence the complexity of the d-Cut problem. Graphs with specific connectivity characteristics may exhibit different complexity behaviors for d=1 and d≥2.
Clique structure: Examining the presence of cliques or clique-like structures in graphs could be another avenue for exploring complexity differences. Graphs with certain clique configurations or clique-free properties may show varied complexity outcomes for different values of d.
Graph coloring: Exploring connections between the d-Cut problem and graph coloring, such as vertex coloring or edge coloring, could provide additional insights. The interplay between d-Cut constraints and coloring requirements could lead to unique complexity scenarios.

By considering these and other structural parameters, we can broaden the scope of the d-Cut problem analysis and uncover new complexity distinctions between the cases of d=1 and d≥2.

Are there any connections between the d-Cut problem and other graph problems, such as graph partitioning or graph coloring, that could lead to further insights

There are potential connections between the d-Cut problem and other graph problems, such as graph partitioning and graph coloring, that could lead to further insights:

Graph Partitioning: The d-Cut problem inherently involves partitioning the graph into two sets based on the cut edges. Exploring connections with graph partitioning problems, like balanced graph partitioning or minimum cut problems, could provide a deeper understanding of the relationship between d-Cut constraints and partitioning objectives. Techniques from graph partitioning algorithms could be adapted to solve specific instances of the d-Cut problem efficiently.
Graph Coloring: The d-Cut problem can be viewed as a specialized form of graph coloring where vertices are colored red or blue based on the d-neighbors constraint. Investigating similarities and differences between d-Cut and graph coloring problems, such as vertex coloring or list coloring, may reveal common algorithmic approaches or complexity implications. Leveraging insights from graph coloring algorithms could enhance the solution strategies for the d-Cut problem and vice versa.
Network Flow: Considering the d-Cut problem in the context of network flow optimization could offer new perspectives. Formulating d-Cut constraints as flow restrictions and exploring analogies with max-flow min-cut theorems could lead to algorithmic innovations and theoretical advancements. By drawing parallels with network flow problems, we can potentially derive efficient algorithms for solving d-Cut instances in specific graph structures.