C.T. Hoang, F. Maray, S. Olariu, and M. Preissman, A charming class of perfectly orderable graphs, Discrete Math. 102, 67-74, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....b a IUT Mesures Physiques, Universit e du Maine, Avenue O. Messiaen, 72017 Le Mans Cedex, France b Universit e d Orl eans, L.I.F.O. B.P. 6759, 45067 Orl eans Cedex 2, France Abstract This paper generalizes previous works on perfectly orderable graphs by Olariu [11] and by Ho ang and al. [9]. Chv atal de ned a graph to be perfectly orderable [2] if there exists a linear order on its set of vertices such that no induced path abcd with edges ab, bc, cd has both a b and d c. Given a graph G and a vertex v in G such that G v is perfectly orderable, we set some conditions on v for ....

....time, containing quasi brittle graphs, charming graphs and some other classes of perfectly orderable graphs. Keywords: graph, perfectly orderable graph, polynomial time. R esum e Dans ce rapport, nous g en eralisons des travaux ant erieurs d Olariu [11] et de Ho ang, Ma ray, Olariu, Preissmann [9], sur les graphes parfaitement ordonnables. Chv atal a d e ni comme parfaitement ordonnable [2] tout graphe dont il existe un ordre lin eaire sur l ensemble de ses sommets tel qu il n existe pas de chemin induit abcd avec les ar etes ab, bc, cd et ayant a b, d c. Etant donn es un graphe G ....

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C. T. Hoang, F. Maray, S. Olariu and M. Preissmann, "A charming class of perfectly orderable graphs", Discrete Mathematics 102 (1992) 67-74.

....the middle of a P 5 . obtained by adding an edge between the end vertices of a P 4 in a C 6 . Theorem 4.1. HK] Every graph with no C k for k 5, no P 5 and no D 6 has some vertex not mid P 4 . Moreover, if the graph is not a clique then it has two non adjacent such vertices. 2 Theorem 4.2. [HMOP] Every graph with no C k for k 5, no P 5 and no P 6 has some vertex not end P 5 . 2 It is natural to ask to what extent results of this form might be strengthened: for fixed k, what are the sets of graphs M k and E k for which every induced subgraph has some vertex not mid P k and not end P k ....

....P 5 . There are many other forms of generalization of Dirac s theorem. One involves replacing the property not mid P 3 of the conclusion with a list of possible proper Meyniel Weakly Triangulated Graphs II: A Theorem of Dirac 7 ties, such as not mid P 4 , or not end P 4 ; see [HK] and [HMOP]. Another involves replacing the vertex property with an edge property, for example not the middle edge of a P 4 ; see [CR] and [ES] In closing, we mention one last related result. A graph is chordal bipartite if it is triangulated (or chordal) and bipartite (in other words, bipartite and ....

C.T. Ho`ang, F. Maffray, S. Olariu & M. Preissman, A charming class of perfectly orderable graphs, Discrete Math. 102 (1992) 67-74.

.... f2,5g, f3,5gg, ffl D 6 free P 5 free weakly triangulated graphs [HK] where D 6 is the complement of the domino, the domino being any copy of the graph with vertex set f1,2,3,4,5,6g and edge set ff1,2g,f2,3g,f3,4g,f4,5g,f5,6g,f1,6g,f1,4gg, and ffl P 6 free P 5 free weakly triangulated graphs [HMOP]. In this paper we prove the conjecture in its full generality by presenting a polynomial time algorithm to find a perfect order of any P 5 free weakly triangulated graph. The correctness of our algorithm relies on a result on separating sets, a result on weakly triangulated graphs, and several ....

C.T. Ho`ang, F. Maffray, S. Olariu & M. Preissman, A charming class of perfectly orderable graphs, Discrete Math. 102 (1992) 67-74.

.... Q Q Q Q Q Q Q Q Q Q Q Q T T T T T T T T T T Q Q Q Q Q Q Q Q Q Q Q Q T T T T T T T T T T Fig. 1. A minimal graph for the property every vertex is the middle of a P 5 . non adjacent such vertices. 2 Theorem 4.2. [HMOP] Every graph with no C k for k 5, no P 5 and no P 6 has some vertex not end P 5 . 2 It is natural to ask to what extent results of this form might be strengthened: for fixed k, what are the sets of graphs M k and E k for which every induced subgraph has some vertex not mid P k and not end P k ....

....be strengthened, since for k 4 neither f(M k ) nor f(E k ) contains P 5 . There are many other forms of generalization of Dirac s theorem. One involves replacing the property not mid P 3 of the conclusion with a list of possible properties, such as not mid P 4 , or not end P 4 ; see [HK] and [HMOP]. Another involves replacing the vertex property with an edge property, for example not the middle Meyniel Weakly Triangulated Graphs and a Theorem of Dirac 7 edge of a P 4 ; see [CR] and [ES] Acknowledgements. I thank Derek Corneil and the Department of Computer Science at the University of ....

C.T. Ho`ang, F. Maffray, S. Olariu & M. Preissman, A charming class of perfectly orderable graphs, Discrete Math. 102 (1992) 67-74. On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25, 71-76.

....for solving these problems are still searched and good results are obtained only for particular classes of graphs. One of them is the class of (P 5 ; P 5 ; C 5 ) free graphs, for which efficient algorithms are known due to Ho ang (see [7] for algorithms on perfectly orderable graphs, and [8] for a proof that (P 5 ; P 5 ; C 5 ) free graphs are perfectly orderable) These algorithms essentially use the fact that no C 5 is induced in the graph, therefore they cannot be easily extended to (P 5 ; P 5 ) free graphs. But we can consider applying these algorithms for some suitable parts of ....

C.T. Ho`ang, F. Maffray, S. Olariu, M. Preissmann, A charming class of perfectly orderable graphs, Discr. Math. 102 (1992), 67-74.

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C.T. Hoang, F. Maray, S. Olariu, and M. Preissman, A charming class of perfectly orderable graphs, Discrete Math. 102, 67-74, 1992.

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