Complexity and algorithms for matching cut problems in graphs without long induced paths and cycles

TL;DR

This work advances the complexity landscape of matching-cut problems by showing that pmc is NP-complete in $P_{14}$-free graphs while MC and DPM are also hard in the same class; it further provides ETH-based subexponential lower bounds for these problems on $P_{14}$-free $8$-chordal graphs. On the algorithmic side, the authors prove polynomial-time solvability of DPM and pmc in $4$-chordal graphs, using forcing-rule techniques for DPM and a novel BFS-level to 2-SAT reduction for pmc. Together, these results unify hardness and tractability across chordal and $P_t$-free graph families and address open questions about the boundary between NP-hardness and tractability for PMC in $k$-chordal graphs.

Abstract

In a graph, a (perfect) matching cut is an edge cut that is a (perfect) matching. Matching Cut (MC), respectively, Perfect Matching Cut (PMC), is the problem of deciding whether a given graph has a matching cut, respectively, a perfect matching cut. The Disconnected Perfect Matching problem (DPM) is to decide if a graph has a perfect matching that contains a matching cut. Solving an open problem posed in [Lucke, Paulusma, Ries (ISAAC 2022, Algorithmica 2023)], we show that PMC is NP-complete in graphs without induced 14-vertex path $P_{14}$. Our reduction also works simultaneously for MC and DPM, improving the previous hardness results of MC on $P_{15}$-free graphs and of DPM on $P_{19}$-free graphs to $P_{14}$-free graphs for both problems. Actually, we prove a slightly stronger result: within $P_{14}$-free 8-chordal graphs (graphs without chordless cycles of length at least 9), it is hard to distinguish between those without matching cuts (respectively, perfect matching cuts, disconnected perfect matchings) and those in which every matching cut is a perfect matching cut. Moreover, assuming the Exponential Time Hypothesis, none of these problems can be solved in $2^{o(n)}$ time for $n$-vertex $P_{14}$-free 8-chordal graphs. On the positive side, we show that, as for MC [Moshi (JGT 1989)], DPM and PMC are polynomially solvable when restricted to 4-chordal graphs. Together with the negative results, this partly answers an open question on the complexity of PMC in $k$-chordal graphs asked in [Le, Telle (WG 2021, TCS 2022) & Lucke, Paulusma, Ries (MFCS 2023, TCS 2024)].

Figures (6)

• Figure 1: Some example graphs; bold edges indicate a matching in question. (a): a matching cut. (b): a perfect matching that is neither a matching cut nor a disconnected perfect matching; this graph has no disconnected perfect matching, hence no perfect matching cut. (c): a perfect matching cut, hence a disconnected perfect matching. (d): a disconnected perfect matching that is not a perfect matching cut.
• Figure 2: From left to right: the graph $H$ with a degree-3 vertex $v$, the graph $G(H;v)$, the graph $G(H;v)$ where $H$ is the Petersen graph, and the graph $G(H;v)$ where $H$ is the Heggernes-Telle graph.
• Figure 3: The gadget $G(C_j)$.
• Figure 4: The graph $G$ from the formula $\phi$ with three clauses $C_1=\{x,y,z\}$, $C_2=\{u,z,y\}$ and $C_3=\{z,v,w\}$. The 6 flax vertices $c_1, c_2, c_3, c_1', c_2', c_3'$ and the 9 teal vertices $a_{11}$, $a_{12}$, $a_{13}$, $a_{21}$, $a_{22}$, $a_{23}$, $a_{31}$, $a_{32}$, $a_{33}$ form the clique $F$ and $T$, respectively.
• Figure 5: The perfect matching cut $(X,Y)$ of the example graph $G$ in Fig. \ref{['fig:P14freeExample1']} given the assignment $y=v= \text{True}$, $x=z = u=w= \text{False}$. $X$ and $Y$ consist of the flax and teal vertices, respectively.

Theorems & Definitions (20)

• Theorem 1: LuckePR23
• Theorem 2: Moshi89
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3