Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization

A matching cut is a partition of the vertex set graph into two sets and B such that each has at most one neighbor in other side cut. The Matching Cut problem asks whether cut, been intensively studied literature. Motivated by question posed Komusiewicz et al. [Discrete Applied Mathematics, 2020], we introduce natural generalization this problem, which call d -Cut: for positive integer d, d-cut bipartition neighbors across We generalize (and some cases, improve) number results problem. Namely, begin with an NP-hardness reduction -Cut on $$(2d+2)$$ -regular graphs polynomial algorithm maximum degree $$d+2$$ . bound hardness result unlikely to be improved, as it would disprove long-standing conjecture context internal partitions. then give FPT algorithms several parameters: edges crossing treewidth, distance cluster, co-cluster. In particular, treewidth improves upon running time best known Cut. Our main technical contribution, building techniques [DAM, kernel every parameterized deletion input cluster graph. also rule out existence kernels when parameterizing simultaneously degree. Finally, provide exact exponential slightly faster than naive brute force approach $$\mathcal {O}^*\!\left( 2^n\right)$$