🤖 AI Summary
This paper studies the $d$-Cut problem: deciding whether a graph admits an edge cut such that each vertex has at most $d$ neighbors on the opposite side of the cut (reducing to Matching Cut when $d=1$). We conduct the first systematic complexity classification for two structured graph classes: graphs of bounded diameter/radius and $H$-free graphs. Our approach integrates combinatorial graph theory, modular decomposition, forbidden-subgraph characterizations, polynomial-time algorithm design, and refined NP-hardness reductions. Main contributions are: (1) a complete complexity dichotomy for graphs parameterized by diameter or radius; (2) polynomial-time algorithms for $P_5$-free, $(P_3+P_4)$-free, and diameter-2 graphs; and (3) an almost complete dichotomy for $H$-free graphs, revealing a fundamental complexity shift between $d=1$ and $dgeq 2$.
📝 Abstract
The $d$-Cut problem is to decide if a graph has an edge cut such that each vertex has at most $d$ neighbours at the opposite side of the cut. If $d=1$, we obtain the intensively studied Matching Cut problem. The $d$-Cut problem has been studied as well, but a systematic study for special graph classes was lacking. We initiate such a study and consider classes of bounded diameter, bounded radius and $H$-free graphs. We prove that for all $dgeq 2$, $d$-Cut is polynomial-time solvable for graphs of diameter $2$, $(P_3+P_4)$-free graphs and $P_5$-free graphs. These results extend known results for $d=1$. However, we also prove several NP-hardness results for $d$-Cut that contrast known polynomial-time results for $d=1$. Our results lead to full dichotomies for bounded diameter and bounded radius and to almost-complete dichotomies for $H$-free graphs.