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A New Characterization of \(P_k\)-free Graphs

  • Eglantine CambyEmail author
  • Oliver Schaudt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

Abstract

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] [12].

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 \(P_k\)-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-Colora bility, an NP-complete problem in general, can be solved in polynomial time for hypergraphs whose vertex-hyperedge incidence graph is \(P_7\)-free.

Keywords

\(P_k\)-free graph Connected domination Computational complexity 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité Libre de BruxellesBrusselsBelgium
  2. 2.Institut für Informatik, Universität zu KölnKölnGermany

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