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 recently posed in [Lucke, Paulusma, Ries (ISAAC 2022), and
Feghali, Lucke, Paulusma, Ries (arXiv:2212.12317)], we show that PMC is
NP-complete in graphs without induced 14-vertex path P14β. Our reduction
also works simultaneously for MC and DPM, improving the previous hardness
results of MC on P19β-free graphs and of DPM on P23β-free graphs to
P14β-free graphs for both problems.
Actually, we prove a slightly stronger result: within P14β-free graphs,
it is hard to distinguish between (i) those without matching cuts and those in
which every matching cut is a perfect matching cut, (ii) those without perfect
matching cuts and those in which every matching cut is a perfect matching cut,
and (iii) those without 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 time 2o(n) for n-vertex P14β-free input graphs.
We also consider the problems in graphs without long induced cycles. It is
known that MC is polynomially solvable in graphs without induced cycles of
length at least 5 [Moshi (JGT 1989)]. We point out that the same holds for DPM.Comment: To appear in the proceedings of WG 202