Colouring Probe $H$-Free Graphs
Daniël Paulusma, Johannes Rauch, Erik Jan van Leeuwen
TL;DR
The paper investigates Colouring and $k$-Colouring on partitioned probe $H$-free graphs, where probes $P$ are known and non-probes $N$ form an independent set whose edges may be added to enforce $H$-freeness. It establishes a full dichotomy for Colouring on partitioned probe $H$-free graphs: polynomial-time when $H\subseteq_i P_4$ and NP-hard otherwise, and a $3$-Colouring dichotomy on partitioned probe $P_t$-free graphs: polynomial for $t\le 5$ and NP-complete for $t\ge 6$, highlighting a sharp complexity shift due to incomplete edge information. The methods combine propagation-based colour-extension, a 2-SAT reduction for fixed colour lists, and intricate structural decompositions, with hardness provided by gadget-based reductions from Exact 3-Cover and related precolouring-extension problems. The results illuminate how partial knowledge of edges alters tractability relative to classical $H$-free graphs and point to rich future directions in recognition, higher-colour Colouring, and broader probe-graph classes.
Abstract
The NP-complete problems Colouring and k-Colouring $(k\geq 3$) are well studied on $H$-free graphs, i.e., graphs that do not contain some fixed graph $H$ as an induced subgraph. We research to what extent the known polynomial-time algorithms for $H$-free graphs can be generalized if we only know some of the edges of the input graph. We do this by considering the classical probe graph model introduced in the early nineties. 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 $G+F$ is $H$-free for some edge set $F\subseteq \binom{N}{2}$. We first fully classify the complexity of Colouring on partitioned probe $H$-free graphs and show that this dichotomy is different from the known dichotomy of Colouring for $H$-free graphs. Our main result is a dichotomy of $3$-Colouring for partitioned probe $P_t$-free graphs: we prove that the problem is polynomial-time solvable if $t\leq 5$ but NP-complete if $t\geq 6$. In contrast, $3$-Colouring on $P_t$-free graphs is known to be polynomial-time solvable if $t\leq 7$ and quasi polynomial-time solvable for $t\geq 8$.