Abstract
A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [Discrete Applied Mathematics, 2020], we introduce a natural generalization of this problem, which we call d -Cut: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the Matching Cut problem. Namely, we begin with an NP-hardness reduction for d -Cut on $$(2d+2)$$ -regular graphs and a polynomial algorithm for graphs of maximum degree at most $$d+2$$ . The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for Matching Cut. Our main technical contribution, building on the techniques of Komusiewicz et al. [DAM, 2020], is a polynomial kernel for d -Cut for every positive integer d, parameterized by the vertex deletion distance of the input graph to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time $$\mathcal {O}^*\!\left( 2^n\right)$$ .
Highlights
A cut of a graph G = (V, E) is a bipartition of its vertex set V (G) into two non-empty sets, denoted by (A, B)
For a positive integer d ≥ 1, a d-cut is a a cut (A, B) such that each vertex has at most d neighbors across the partition, that is, every vertex in A has at most d neighbors in B, and vice-versa
Afterwards, we present a dynamic programming algorithm for d-Cut parameterized by treewidth running in time O∗ 2tw(G)(d + 1)2tw(G) ; in particular, for d = 1 this algorithm runs in time O∗ 8tw(G) and improves the one given by Aravind et al [1] for Matching Cut, which runs in O∗ 12tw(G) time
Summary
A problem closely related to d-Cut is that of Internal Partition, first studied by Thomassen [28] In this problem, we seek a bipartition of the vertices of an input graph such that every vertex has at least as many neighbors in its own part as in the other part. By employing the cross-composition framework of Bodlaender et al [4] and using a reduction similar to the one in [18], we show that, unless NP ⊆ coNP/poly, there is no polynomial kernel for d-Cut parameterized simultaneously by the number of crossing edges, the maximum degree, and the treewidth of the input graph. We give an FPT algorithm parameterized by the distance to co-cluster, denoted by dc(G) These results imply the existence of a polynomial kernel for d-Cut parameterized by the vertex cover number τ (G).
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