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 Pkfree graphs, including alternative characterizations. Let G be a connected Pk-free graph, k ≥ 4. We show that G admits a connected dominating set that induces either a Pk−2-free graph or a graph isomorphic to Pk−2. Surprisingly, it turns out that every minimum connected dominating set of G has this property. This yields a new characterization for Pkfree graphs: a graph G is Pk-free if and only if each connected induced subgraph ofG has a connected dominating set that induces either a Pk−2free graph, or a graph 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. In the second part of the paper, we present an efficient algorithm that, given a connected graph G, 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 Pk−2free or isomorphic to Pk−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 graphs is P7-free. keywords: Pk-free graph, connected domination, computational complexity. MSC: 05C69, 05C75, 05C38.