Finding $d$-Cuts in Probe $H$-Free Graphs

TL;DR

This work studies the $d$-Cut problem on partitioned probe $H$-free graphs, integrating partial edge information via probes and certifiable edges so that $G+F$ is $H$-free. The authors leverage a red-blue $d$-colouring framework and colour-processing rules to reduce the search space to a polynomial number of branches, enabling polynomial-time algorithms for several cases. They establish complete dichotomies: for partitioned probe graphs, $1$-Cut, Perfect Matching Cut, and Maximum Matching Cut are polynomial when $H\subseteq_i sP_1+P_4$ for some $s\ge0$, while for $d\ge2$, $d$-Cut is polynomial when $H\subseteq_i P_1+P_4$; otherwise the problems are NP-complete. The paper also proves NP-hardness for several restricted classes (e.g., probe $2P_2$-free, probe $K_{1,3}$-free, and probe $4P_1$-free), delineating sharp tractability boundaries in the probe-graph model and outlining future directions for broader problem classes.

Abstract

For an integer $d\geq 1$, the $d$-Cut problem is that of deciding whether a graph has an edge cut in which each vertex is adjacent to at most $d$ vertices on the opposite side of the cut. The $1$-Cut problem is the well-known Matching Cut problem. The $d$-Cut problem has been extensively studied for $H$-free graphs. We extend these results to the probe graph model, where we do not know all the edges of the input graph. For a graph $H$, a partitioned probe $H$-free graph $(G,P,N)$ consists of a graph $G=(V,E)$, together with a set $P\subseteq V$ of probes and an independent set $N=V\setminus P$ of non-probes such that we can change $G$ into an $H$-free graph by adding zero or more edges between vertices in $N$. For every graph $H$ and every integer $d\geq 1$, we completely determine the complexity of $d$-Cut on partitioned probe $H$-free graphs.

Figures (9)

• Figure 1: A graph $G$ with a set $P$ of probes. The set $F$ is the set of dashed edges. The blue-red colouring corresponds to a $2$-cut of $G$ and a $3$-cut in $G+F$.
• Figure 2: The graphs $sP_1+P_4$, $K_{1,3}$ and $H^*_i$, from left to right.
• Figure 3: Illustration of Case 11.2. Lines represent edges and dashed lines represent non-edges. Note that $x'$ and $y'$ may not belong to $X_1$ and $X_2$, respectively, but this is not relevant (it only matters that they are not adjacent to $b$).
• Figure 4: Illustration of the types described in Case 1.7.3, specified by vertex label. A (dashed) purple line between a vertex $v\in N$ and some $C_i$ represents that $v$ is (anti-)complete to $C_i$. A black (dashed) edge between a vertex $v\in N$ and a vertex in $P$ represents a (non)-edge.
• Figure 5: A $P$-dominating pair $\{u,v\}$.

Theorems & Definitions (23)

• theorem 1.1: AELPSBo09Ch84FLPR23LL23LMO25LMPS24LPR22LPR23aMo89
• theorem 1.2: FLPR23LL23LT22LPR23a
• theorem 1.3: LPR24
• theorem 1.4
• lemma thmcounterlemma: LMPS24
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• theorem 1.6