Finding d-Cuts in Claw-Free Graphs

초록

The Matching Cut problem is to decide if the vertex set of a connected graph can be partitioned into two non-empty sets B and R such that the edges between B and R form a matching, that is, every vertex in B has at most one neighbour in R, and vice versa. If for some integer d ≥ 1, we allow every vertex in B to have at most d neighbours in R, and vice versa, we obtain the more general problem d-Cut. It is known that d-Cut is NP-complete for every d ≥ 1. However, for claw-free graphs, it is only known that d-Cut is polynomial-time solvable for d = 1 and NP-complete for d ≥ 3. We resolve the missing case d = 2 by proving NP-completeness. This follows from our more general study, in which we also bound the maximum degree. That is, we prove that for every d ≥ 2, d-Cut, restricted to claw-free graphs of maximum degree p, is constant-time solvable if p ≤ 2d + 1 and NP-complete if p ≥ 2d + 3. Moreover, in the former case, we can find a d-cut in linear time. We also show how our positive results for claw-free graphs can be generalized to S1t,ℓ-free graphs where S1t,ℓ is the graph obtained from a star on t + 2 vertices by subdividing one of its edges exactly ℓ times. © 2025 Jungho Ahn, Tala Eagling-Vose, Felicia Lucke, Daniël Paulusma, and Siani Smith.

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