Mathematics > Combinatorics
[Submitted on 17 Apr 2024 (v1), last revised 29 Oct 2024 (this version, v2)]
Title:Finding $d$-Cuts in Graphs of Bounded Diameter, Graphs of Bounded Radius and $H$-Free Graphs
View PDF HTML (experimental)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.
Submission history
From: Felicia Lucke [view email][v1] Wed, 17 Apr 2024 13:51:20 UTC (114 KB)
[v2] Tue, 29 Oct 2024 10:08:30 UTC (111 KB)
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