Finding Minimum Matching Cuts in $H$-free Graphs and Graphs of Bounded Radius and Diameter

TL;DR

This paper studies Minimum Matching Cut, the problem of finding a matching that is also an edge cut with minimum size, comparing it to the classical Matching Cut. Using a red-blue colouring framework and safe propagation rules, it obtains polynomial-time algorithms for $P_7$-free, $S_{1,1,2}$-free, and $(P_6+P_4)$-free graphs, and proves NP-hardness for $3P_3$-free graphs, separating the two problems on certain $H$-free classes. It also derives dichotomies for Minimum Matching Cut and for Matching Cut on graphs of bounded radius/diameter, including bipartite variants. Together, the results complete several open cases in the $H$-free dichotomy landscape and illuminate how structural graph restrictions impact the tractability of matching-cut problems.

Abstract

A matching cut is a matching that is also an edge cut. In the problem Minimum Matching Cut, we ask for a matching cut with the minimum number of edges in the matching. We give polynomial-time algorithms for $P_7$-free, $S_{1,1,2}$-free and $(P_6 + P_4)$-free graphs, which also solve several open cases for the well-studied problem Matching Cut. In addition, we show NP-hardness for $3P_3$-free graphs, implying that Minimum Matching Cut and Matching Cut differ in complexity on certain graph classes. We also give complexity dichotomies for both general and bipartite graphs of bounded radius and diameter.

Figures (2)

• Figure 1: A perfect matching in $G'$ (left) and the corresponding minimum valid red-blue colouring.
• Figure 2: The connected components $C$ and $C'$ of $G[Z]$.

Theorems & Definitions (25)

• theorem thmcountertheorem
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