Finding d-Cuts in Claw-free Graphs

TL;DR

The paper advances the complexity classification of the $d$-Cut problem on claw-free graphs, resolving the open case $d=2$ by proving NP-completeness and establishing a sharp degree-based boundary: linear-time, constructive solvability for maximum degree $p \le 2d+1$ and NP-completeness for $p \ge 2d+3$, with extensions to $S_{1^t,\ell}$-free graphs. It also provides a strong existence result for large, $S_{1^t,\ell}$-free claw-free graphs and introduces tight lower-bound constructions showing no $d$-cut at degree $2d+2$. The NP-hardness proof uses a careful reduction from NAE $3$-Sat $0$-$1$ with gadgets $H_{d,k,r}$ and $F_\ell$, linking red-blue $d$-colorings to NAE-satisfying assignments that use both truth values. These results refine the boundary between tractable and intractable instances of $d$-Cut in claw-free and related $H$-free classes, and they extend the positive results to broader graph families through $S_{1^t,\ell}$-free generalizations.

Abstract

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 $d\geq 1$, we allow every neighbour 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 $d\geq 1$. However, for claw-free graphs, it is only known that $d$-Cut is polynomial-time solvable for $d=1$ and NP-complete for $d\geq 3$. 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 $d\geq 2$, $d$-Cut, restricted to claw-free graphs of maximum degree $p$, is constant-time solvable if $p\leq 2d+1$ and NP-complete if $p\geq 2d+3$. 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 $S_{1^t,l}$-free graphs where $S_{1^t,l}$ is the graph obtained from a star on $t+2$ vertices by subdividing one of its edges exactly $l$ times.

Figures (7)

• Figure 1: Left: a graph with a matching cut ($1$-cut). Right: a graph with a $3$-cut but no $d$-cut for $d\leq 2$. Figure taken from LMPS24.
• Figure 2: The graph $S_{1^6,4}$.
• Figure 3: A chain of diamonds has no $1$-cut (the diamond is the graph $K_4-e$).
• Figure 4: An example of the procedure in the proof of Lemma \ref{['lem:cubic']} if ${d=2}$. Left: every vertex in $S$ has at most $2$ neighbours outside of $S$. The edges in $\delta(S)$ are highlighted in black. Right: The red-blue $d$-colouring we obtain from the procedure.
• Figure 5: The graph constructed in Theorem \ref{['thm:examples']} for ${k=3}$. Note that for each $i\in[3]$, ${A_i\cup B_i}$ is a clique and $v_i$ is complete to $B_i\cup A_{i+1}$, where $A_4:=A_1$,

Theorems & Definitions (10)

• Theorem 1: Gomes and Sau GS21
• Theorem 2: Gomes and Sau GS21
• Theorem 3
• Theorem 4
• Lemma 9
• Theorem 10
• Theorem 11
• Proposition 12: Eagling-Vose et al. EMPS25
• Lemma 13
• Theorem 14