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arXiv · 2404.11389

Finding d-Cuts in Graphs of Bounded Diameter, Graphs of Bounded Radius and H-Free Graphs

Felicia Lucke·Ali Momeni·Daniël Paulusma·Siani Smith
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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 $d\geq 2$, 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.

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Felicia Lucke, Ali Momeni, Daniël Paulusma, Siani Smith. 2025-10-06. Finding d-Cuts in Graphs of Bounded Diameter, Graphs of Bounded Radius and H-Free Graphs. https://arxiv.org/abs/2404.11389

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