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

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 $\mathrm{NP}$-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 $2^{o\left(n\right)}$ 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)].