Edge and vertex intersection of paths in a tree.

*(English)*Zbl 0568.05023The path graph of a tree T is a graph whose vertex set is the set \({\mathcal P}\) of nontrivial simple paths in T and in which two vertices are adjacent if and only if they (as paths) have a common vertex. Similarly the edge intersection graph of paths (shortly EPT graph) of a tree T is defined; its vertex set is again \({\mathcal P}\) and two vertices are adjacent in it if and only if they have a common edge. First a theorem on maximal cliques of an EPT graph is proved. Further the authors present a characterization of graphs which are simultaneously path graphs and EPT graphs of some trees. At the end it is proved that recognizing whether a given graph is an EPT graph is an NP-complete problem.

Reviewer: B.Zelinka

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\textit{M. C. Golumbic} and \textit{R. E. Jamison}, Discrete Math. 55, 151--159 (1985; Zbl 0568.05023)

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##### References:

[1] | Berge, C., Graphs and hypergraphs, (1973), North-Holland Amsterdam · Zbl 0483.05029 |

[2] | Gavril, F., A recognition algorithm for the intersection graphs of paths in trees, Discrete math., 23, 211-227, (1978) · Zbl 0398.05060 |

[3] | Golumbic, M.C., Algorithmic graph theory and perfect graphs, (1980), Academic Press New York · Zbl 0541.05054 |

[4] | Golumbic, M.C.; Jamison, R.E., The edge intersection graphs of paths in a tree, J. combin. theory ser. B, 37, (1985) · Zbl 0537.05063 |

[5] | Lobb, W.A., Perfect graphs from paths in trees, (), presented at |

[6] | Renz, P.L., Intersection representations of graphs by arcs, Pacific J. math., 34, 501-510, (1970) · Zbl 0191.55103 |

[7] | Shannon, C.E., A theorem on coloring the lines of a network, J. math. phys., 28, 148-151, (1949) · Zbl 0032.43203 |

[8] | Syslo, M.M., On characterizations of cycle graphs, Problémes combinatoires et théorie des graphes, 395-398, (1978), Colloq. CNRS, Orsay 1976 · Zbl 0412.05057 |

[9] | Syslo, M.M., On characterizations of cycle graphs and on other families of intersection graphs, () · Zbl 0418.05047 |

[10] | Syslo, M.M., Triangulated edge intersection graphs of paths in a tree, Discrete math., 55, 217-220, (1985) · Zbl 0569.05045 |

[11] | Tarjan, R.E., Decomposition by clique separators, Discrete math., 55, 221-232, (1985) · Zbl 0572.05039 |

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