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Matching Cut and Variants on Bipartite Graphs of Bounded Radius and Diameter

Felicia Lucke

TL;DR

The paper studies the computational complexity of Matching Cut and its variants on bipartite graphs with bounded radius and diameter, introducing a red-blue colouring transformation that maps each problem to a corresponding colouring constraint. It derives complexity dichotomies for the $d$-Cut and Maximum Matching Cut problems under radius/diameter bounds, provides a polynomial-time algorithm for Perfect Matching Cut on bipartite graphs with diameter $3$, and proves NP-hardness for radius $4$, while resolving one open case for Disconnected Perfect Matching. The approach unifies several problems through the coloring framework and yields a comprehensive complexity map (including a tabular overview) for these problems in restricted graph classes. This advances the understanding of how structural graph parameters constrain the tractability of matching-based cut problems and informs algorithm design for network reliability and partitioning tasks.

Abstract

In the Matching Cut problem we ask whether a graph $G$ has a matching cut, that is, a matching which is also an edge cut of $G$. We consider the variants Perfect Matching Cut and Disconnected Perfect Matching where we ask whether there exists a matching cut equal to, respectively contained in, a perfect matching. Further, in the problem Maximum Matching Cut we ask for a matching cut with a maximum number of edges. The last problem we consider is $d$-Cut where we ask for an edge cut where each vertex is incident to at most $d$ edges in the cut. We investigate the computational complexity of these problems on bipartite graphs of bounded radius and diameter. Our results extend known results for Matching Cut and Disconnected Perfect Matching. We give complexity dichotomies for $d$-Cut and Maximum Matching Cut and solve one of two open cases for Disconnected Perfect Matching. For Perfect Matching Cut we give the first hardness result for bipartite graphs of bounded radius and diameter and extend the known polynomial cases.

Matching Cut and Variants on Bipartite Graphs of Bounded Radius and Diameter

TL;DR

The paper studies the computational complexity of Matching Cut and its variants on bipartite graphs with bounded radius and diameter, introducing a red-blue colouring transformation that maps each problem to a corresponding colouring constraint. It derives complexity dichotomies for the -Cut and Maximum Matching Cut problems under radius/diameter bounds, provides a polynomial-time algorithm for Perfect Matching Cut on bipartite graphs with diameter , and proves NP-hardness for radius , while resolving one open case for Disconnected Perfect Matching. The approach unifies several problems through the coloring framework and yields a comprehensive complexity map (including a tabular overview) for these problems in restricted graph classes. This advances the understanding of how structural graph parameters constrain the tractability of matching-based cut problems and informs algorithm design for network reliability and partitioning tasks.

Abstract

In the Matching Cut problem we ask whether a graph has a matching cut, that is, a matching which is also an edge cut of . We consider the variants Perfect Matching Cut and Disconnected Perfect Matching where we ask whether there exists a matching cut equal to, respectively contained in, a perfect matching. Further, in the problem Maximum Matching Cut we ask for a matching cut with a maximum number of edges. The last problem we consider is -Cut where we ask for an edge cut where each vertex is incident to at most edges in the cut. We investigate the computational complexity of these problems on bipartite graphs of bounded radius and diameter. Our results extend known results for Matching Cut and Disconnected Perfect Matching. We give complexity dichotomies for -Cut and Maximum Matching Cut and solve one of two open cases for Disconnected Perfect Matching. For Perfect Matching Cut we give the first hardness result for bipartite graphs of bounded radius and diameter and extend the known polynomial cases.
Paper Structure (2 sections, 1 figure, 1 table)

This paper contains 2 sections, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries

Figures (1)

  • Figure 1: A valid red-blue colouring of value $4$ (left), a perfect extendable red-blue colouring of value $1$ (middle) and a perfect red-blue colouring (right).