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

TL;DR

This work studies the d-Cut problem, a generalization of Matching Cut, focusing on graph classes with bounded diameter, bounded radius, and H-free graphs. It shows that for all $d\ge 2$, d-Cut is polynomial-time solvable on graphs of diameter 2, and on $(P_3+P_4)$-free and $P_5$-free graphs, while also proving several NP-hardness results that yield (almost) complete dichotomies for these restricted classes. A key methodological contribution is the red-blue $d$-coloring framework and domination-based colouring techniques, including the precoloured pair concept and colour-processing rules, as well as a general lemma linking $H$-free and $(H+P_1)$-free cases. The results collectively extend known $d=1$ classifications, highlight complexity jumps between $d=1$ and $d\ge 2$ for certain graph families, and establish a near-complete global landscape with only a few open $H$-free cases remaining. These insights advance understanding of how structural graph restrictions influence the tractability of edge-cut problems and inform future work on the remaining open classifications.

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.

Figures (8)

• Figure 1: Left: a graph with a matching cut (i.e., a $1$-cut). Middle: a graph with a $3$-cut but no $d$-cut for $d\leq 2$. Right: the graph $H_i^*$.
• Figure 2: Cases 1 (left) and 2 (right) in the proof of Theorem \ref{['t-diam']}.
• Figure 3: The graph $G$ in the blue phase (left) and in the red phase: Case 2 (right).
• Figure 4: Illustration of Case 1. As $Q=u_1r_1rr_2$ is induced, the displayed $P_3+P_4$ cannot be induced. White vertices indicate uncoloured vertices.
• Figure 5: Illustration of Case 2. As $J = rr_1u$ is induced, the displayed $P_3+P_4$ cannot be induced. White vertices indicate uncoloured vertices.

Theorems & Definitions (31)

• theorem 1.1: LL19LPR22
• theorem 1.2: Bo09Ch84FLPR25LL23LPR22LPR23aMo89
• theorem 1.3
• theorem 1.4
• theorem 1.5
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• theorem 1.7