A new characterization of $P_k$-free graphs

The class of graphs that do not contain an induced path on $k$ vertices, $Pk$-free graphs, plays a prominent role in algorithmic graph theory. This motivates the search for special structural properties of $Pk$-free graphs, including alternative characterizations. Let $G$ be a connected $Pk$-free graph, $k \ge 4$. We show that $G$ admits a connected dominating set whose induced subgraph is either $P{k-2}$-free, or isomorphic to $P{k-2}$. Surprisingly, it turns out that every minimum connected dominating set of $G$ has this property. This yields a new characterization for $Pk$-free graphs: a graph $G$ is $Pk$-free if and only if each connected induced subgraph of $G$ has a connected dominating set whose induced subgraph is either $P{k-2}$-free, or isomorphic to $Ck$. This improves and generalizes several previous results; the particular case of $k=7$ solves a problem posed by van 't Hof and Paulusma [A new characterization of $P6$-free graphs, COCOON 2008]. In the second part of the paper, we present an efficient algorithm that, given a connected graph $G$ on $n$ vertices and $m$ edges, computes a connected dominating set $X$ of $G$ with the following property: for the minimum $k$ such that $G$ is $Pk$-free, the subgraph induced by $X$ is $P{k-2}$-free or isomorphic to $P{k-2}$. As an application our results, we prove that Hypergraph 2-Colorability, an NP-complete problem in general, can be solved in polynomial time for hypergraphs whose vertex-hyperedge incidence graph is $P7$-free.