A new characterization of $P_k$-free graphs

The class of graphs that do not contain an induced path on $k$ vertices, $P_k$-free graphs, plays a prominent role in algorithmic graph theory. This motivates the search for special structural properties of $P_k$-free graphs, including alternative characterizations. Let $G$ be a connected $P_k$-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 $P_k$-free graphs: a graph $G$ is $P_k$-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 $C_k$. 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 $P_6$-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 $P_k$-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 $P_7$-free.

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