A faster algorithm to recognize undirected path graphs.

*(English)*Zbl 0770.68096Summary: Let \(\mathcal F\) be a finite family of nonempty sets. The undirected graph \(G\) is called the intersection graph of \(\mathcal F\) if there is a bijection between the members of \(\mathcal F\) and the vertices of \(G\) such that any two sets \(F_ i\) and \(F_ j\) (for \(i \neq j\)) have a nonempty intersection if and only if the corresponding vertices are adjacent. We study intersection graphs where \(\mathcal F\) is a family of undirected paths in an unrooted, undirected tree; these graphs are called (undirected) path graphs. They constitute a proper subclass of the chordal graphs. F. Gavril [Discrete Math. 23, 211-227 (1978; Zbl 0398.05060)] gave the first polynomial time algorithm to recognize undirected path graphs; his algorithm runs in time \(O(n^ 4)\), where \(n\) is the number of vertices. The topic of this paper is a new recognition algorithm that runs in time \(O(mn)\), where \(m\) is the number of edges.

##### MSC:

68R10 | Graph theory (including graph drawing) in computer science |

68Q25 | Analysis of algorithms and problem complexity |

05C05 | Trees |

05C75 | Structural characterization of families of graphs |

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\textit{A. A. Schäffer}, Discrete Appl. Math. 43, No. 3, 261--295 (1993; Zbl 0770.68096)

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