Dichotomies for Maximum Matching Cut: $H$-Freeness, Bounded Diameter, Bounded Radius

TL;DR

This work studies Maximum Matching Cut, the optimization version of the classical Matching Cut problem, and relates it to Perfect and Disconnected variants. It develops a unified red-blue colouring framework to derive polynomial-time algorithms for graphs with diameter at most $2$, for $P_6$-free graphs, and for $H$-free classes with $H\subseteq_i sP_2+P_6$, while proving NP-hardness in carefully constructed cases such as diameter $3$ and radius $2$, and in subcubic line graphs. The authors establish complete dichotomies across graph diameter, radius, and $H$-free classes, and extend the techniques to Maximum Disconnected Perfect Matching, yielding parallel tractable/intractable boundaries and new results for Disconnected Perfect Matching. Collectively, the results sharpen our understanding of how maximizing a matching cut interacts with structural graph restrictions and have implications for related optimization variants and classification questions in graph algorithms.

Abstract

The (Perfect) Matching Cut problem is to decide if a graph $G$ has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of $G$. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and $H$-free graphs. A disconnected perfect matching of a graph $G$ is a perfect matching that contains a matching cut of $G$. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for $(P_6+sP_2)$-free graphs for every $s\geq 0$, extending a known result for $P_5$-free graphs (Bouquet and Picouleau, 2020).

Figures (6)

• Figure 1: The graphs $H_1^*$ (left) and $H_i^*$ (right).
• Figure 2: The graph $P_6$ with a matching cut of size $2$ that is not contained in a (disconnected) perfect matching (left), a disconnected perfect matching with a matching cut of size $2$ (middle) and a perfect matching cut (of size $3$) (right). In each figure, thick edges denote matching cut edges.
• Figure 3: An example (from LPR23a) of a red-blue $(S',T',X',Y')$-colouring of a graph with an intermediate $4$-tuple $(S',T',X',Y')$.
• Figure 4: A $U$-saturating matching (left) and the corresponding valid red-blue colouring (right). Note that not every vertex in $X \cup Y$ belongs to $W$.
• Figure 7: A graph $G$ (left) where the tick red edges form a maximum edge cut, and the graph $G'$ (right) from the proof of Theorem \ref{['theo-clawfree']}, where the thick red edges form a maximum matching cut.

Theorems & Definitions (33)

• Theorem 1: LL19LPR22
• Theorem 2: Bo09Ch84FLPR23LL23LPR23aLPR22Mo89
• Theorem 3: HT98LL23LPR23a
• Theorem 4: FLPR23LL23LT22LPR23a
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8: BP
• Theorem 9
• Theorem 10