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

For an integer $d\ge 1$, 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 $d\ge 1$, we completely determine the complexity of $d$-Cut on partitioned probe $H$-free graphs.