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Connectivity, Transitivity and Chromaticity: The Pioneering Work of Bernard Roy in Graph Theory

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 44))

Abstract

We review the work of B. Roy in graph theory and its posterity.

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References

  • Aharoni, R., Ben-Arroyo Hartman, I., and Hoffman, A.J. (1985). Path partitions and packs of acyclic digraphs. Pacific J. of Math. 118:249–259.

    Article  Google Scholar 

  • Bang-Jensen, J., and Gutin, G. (1988). Generalization of tournaments: a survey. J. Graph Theory 28:171–202.

    Article  Google Scholar 

  • Berge, C. (1973). Graphs and Hypergraphs. North Holland, Amsterdam.

    Google Scholar 

  • Berge, C. (1982). Diperfect graphs. Combinatorica 2:213–222.

    Article  Google Scholar 

  • Berge, C. (1984). A property of k-optimal path-partitions. In: Progress in Graph Theory (Waterloo, Ont. 1982). Academic Press, Toronto, 105–108.

    Google Scholar 

  • Berge, C., and Duchet, P. (1984). Strongly perfect graphs. Ann. of Discrete Mathematics 21:57–61.

    Google Scholar 

  • Berge, C., and Ghouila-Houri, A. (1965). Programming, games and transportation networks. Methuen, London.

    Google Scholar 

  • Bermond, J.C., and Thomassen, C. (1981). Cycles in digraphs — A survey. J. of Graph Theory 5:1–43.

    Article  Google Scholar 

  • Bilde, O., and Krarup, J. (1969). A modified cascade algorithm for shortest paths. METRA VIII 231–241.

    Google Scholar 

  • Bondy, J. A. (1976). Disconnected orientations and a conjecture of Las Vergnas. J. London Math. Soc. 2:277–282.

    Article  Google Scholar 

  • Busolini, D.T. (1971). Monochromatic paths and circuits in edge-colored graphs. J. Combinatorial Theory 10:299–300.

    Article  Google Scholar 

  • Cameron, K. (1986). On k-optimum dipath partitions and partial k-colourings of acyclic digraphs. Europ. J. Combinatorics 7:115–118.

    Google Scholar 

  • Camion, P. (1959). Chemins et circuits hamiltoniens des graphes complets. C. R. Acad. Sci. Paris 249:2151–2152.

    Google Scholar 

  • Chvátal, V. (1972). Monochromatic paths in edge-colored graphs. J. of Combinatorial Theory B 13:69–70.

    Article  Google Scholar 

  • Cunninghame-Green, R.A. (1979). Minimax algebra. Lecture Notes in Economics and Mathematical Systems 166, Springer-Verlag, Berlin.

    Book  Google Scholar 

  • Cunninghame-Green, R.A. (1960). Process synchronization in a steelworks — a problem of feasibility. In: Proceed. 2 nd International Conf. on Operational Research, English Univ. Press, 323–328.

    Google Scholar 

  • Cunninghame-Green, R.A. (1962). Describing industrial processes with interference and approximating their steady-state behaviour. Operational Research Quart. 13:95–100.

    Article  Google Scholar 

  • Dantzig, G.B. (1967). All shortest routes in a graph. Proc. International Symp. on Theory of Graphs, Rome, Italy 1966, Paris, Dunod.

    Google Scholar 

  • de Biroille, T. (1534). Private communication.

    Google Scholar 

  • Deo, N., and Pang, Ch.Y. (1984). Shortest-path algorithms: taxonomy and annotation. Networks 14:275–323.

    Article  Google Scholar 

  • Fajtlowicz, S. (1988). On conjectures of Graffiti. Disc. Math. 72:113–118.

    Article  Google Scholar 

  • Fajtlowicz, S. (1999). Written on the wall. Version 9–1999. Regularly updated file available from ~clarson@math.uh.edu

    Google Scholar 

  • Farbey, B.A., Land, A.H., and Murchland, J.D. (1967). The cascade algorithm for finding all shortest distances in a directed graph. Manag. Sci. 14:19–28.

    Article  Google Scholar 

  • Floyd, R.W. (1962). Algorithm 97: shortest path. Communications of the ACM 5:345.

    Article  Google Scholar 

  • Gale, D. (1959). Math Review 20 # 2727 55.00

    Google Scholar 

  • Gallai, T. (1958). Maximum-minimum Sätze über Graphen. Acta Math. Acad. Sci. Hungar. 9:395–434.

    Article  Google Scholar 

  • Gallai, T. (1968). On directed paths and circuits. In: Theory of graphs. Proceed. Colloq. Tihany 1966, Academic press, New York, 115–118.

    Google Scholar 

  • Gallai, T., and Milgram, A.N. (1960). Verallgemeinerung eines Graphen theoretischen Satzes von Rédei. Acta. Sci. Math. 21:181–186.

    Google Scholar 

  • Gallo, G., and Pallottino, S. (1986). Shortest path methods: a unifying approach. Math. Prog. Study 26:38–64.

    Article  Google Scholar 

  • Giffler, R. (1963). Scheduling general production systems using schedule algebra. Naval Res. Logistics Quart. 10:237–255.

    Article  Google Scholar 

  • Ghouila-Houri, A. (1960). Sur l’existence d’un flot ou d’une tension prenant ses valeurs dans un groupe abélien. C. R. Acad. Sci. Paris 250:3931–3933.

    Google Scholar 

  • Goldberg, M., and Moon, J.W. (1972). Cycles in k-strong tournaments. Pacific J.Math. 40:89–96.

    Article  Google Scholar 

  • Goldman, A.J., and Tiwari, P. (1986). Allowable processing orders in the accelerated cascade algorithm. Discrete Appl. Math. 13:213–221.

    Article  Google Scholar 

  • Gondran, M. (1975). Path algebra and algorithms. In: Combinatorial Programming: Methods and Applications. (B. Roy, ed.) NATO Adv. Study Inst. 19, Reidel, Dordrecht, 137–148.

    Chapter  Google Scholar 

  • Gondran, M. (1996a). Analyse MINMAX. C. R. Acad. Sci. Paris 323:1249–1252.

    Google Scholar 

  • Gondran, M. (1996b). Analyse MINPLUS. C. R. Acad. Sci. Paris 323:371–375.

    Google Scholar 

  • Gondran, M., and Minoux, M. (1984). Linear algebra in dioids: a survey of recent results. Ann. of Discrete Mathematics 19:147–164.

    Google Scholar 

  • Gondran, M., and Minoux, M. (1997). Valeurs propres et fonctions propres d’endomorphismes à diagonale dominante en analyse Min-Max. C. R. Acad. Sci. Paris 325:1287–1290.

    Article  Google Scholar 

  • Hammer, P.L., and Nguyen, S. (1977). APOSS. A partial order in the solution space of bivalent problems. In: Modern trends in cybernetics and systems. Proc. Third Internat. congr. Bucharest, 1975. Springer, Berlin, pp. 869–883.

    Google Scholar 

  • Hansen, P. (1976/77). A cascade algorithm for the logical closure of a set of binary relations. Info. Proc. Lett. 5:50–54.

    Article  Google Scholar 

  • Hansen, P., Kuplinsky, J., and de Werra, D. (1997). Mixed graph colorings. Math. Methods of O. R. 45:145–160.

    Article  Google Scholar 

  • Hoffman, A.J., and Winograd, S. (1972). Finding all shortest distances in a directed network. Math. of Numerical Computation, IBM J. Res. Develop. 16:412–414.

    Article  Google Scholar 

  • Hu, T.C. (1967). Revised matrix algorithms for shortest paths. SIAM J. Appl. Math. 15:207–218.

    Article  Google Scholar 

  • Greene, C., and Kleitman, D.J. (1976). The structure of Sperner k-families. J. Combin. Theory A 20:41–68.

    Article  Google Scholar 

  • Gries, D., Martin, A.J., van de Snepscheut, J.L.A., and Udding, J.T. (1989). An algorithm for transitive reduction of an acyclic graph. Sci. Comput. Programming 12:151–155.

    Article  Google Scholar 

  • Klee, V., and Larman, D. (1979). Use of Floyd’s algorithm to find shortest restricted paths. Ann. of Discrete Math. 4:237–249.

    Article  Google Scholar 

  • Land, A.H., and Stairs, S.W. (1967). The extension of the cascade algorithm to large graphs. Manag. Sci. 14:29–33.

    Article  Google Scholar 

  • Li, Hao (1998). A generalization of the Gallai-Roy theorem, preprint, Université de Paris-Sud and Graphs and Combinatorics (to appear).

    Google Scholar 

  • Linial, N. (1978). Covering digraphs by paths. Disc. Math. 23:257–272.

    Google Scholar 

  • Müller, H. (1981). Oriented hypergraphs, stability numbers and chromatic numbers. Disc. Math. 34:319–320.

    Article  Google Scholar 

  • Naur, P. (1994). Proof versus formalization. BIT 34:148–164.

    Article  Google Scholar 

  • Papadimitriou, C, and Sideri, M. (1999). On the Floyd-Warshall algorithm for logic programming. J. Logic Programming 41:129–137.

    Article  Google Scholar 

  • Poel, M., and Zwiers, J. (1993). Layering techniques for development of parallel systems: an algebraic approach, computer aided verification. Lecture notes in Comput. Sci., Springer, Berlin, 663:16–29.

    Article  Google Scholar 

  • Preissmann, M., and de Werra, D. (1985). A note on strong perfectness of graphs. Math. Prog. 32:321–326.

    Article  Google Scholar 

  • Rado, R. (1943). Theorems on linear combinatorial topology and general measure. Ann. of Math. 44:228–270.

    Article  Google Scholar 

  • Robert, P., and Ferland, J. (1968). Généralisation de l’algorithme de Warshall. Revue Française d’Aut. Info, et Rech. Opér. 2:71–85.

    Google Scholar 

  • Rote, G. (1985). A systolic array algorithm for the algebraic path problem (shortest paths; matrix inversion). Computing 34:191–219.

    Article  Google Scholar 

  • Roy, B. (1958). Sur quelques propriétés des graphes fortement connexes. C. R. Acad. Sci. Paris 247:399–401.

    Google Scholar 

  • Roy, B. (1959a). Contribution de la théorie des graphes à l’étude de certains problems linéaires. C. R. Acad. Sci. Paris 248:2437–2439.

    Google Scholar 

  • Roy, B. (1959b). Transitivité et connexité. C. R. Acad. Sci. Paris 249:216–218.

    Google Scholar 

  • Roy, B. (1962). Cheminement et connexité dans les graphes: application aux problems d’ordonnancement. METRA Série Spéciale No 1 (Mai 1962).

    Google Scholar 

  • Roy, B. (1967). Nombre chromatique et plus long chemins d’un graphe. Rev. Info. et Rech. Opér 5:129–132.

    Google Scholar 

  • Roy, B. (1969). Graph partial s-connexe extremum. Revue Roumaine Math. Pures et Appl. XIV, 1355–1368.

    Google Scholar 

  • Roy, B. (1969/70). Algèbre moderne et théorie des graphes. Dunod (Paris, Tome 1:1969, tome 2:1970).

    Google Scholar 

  • Saks, M. (1986). Some sequences associated with combinatorial structures. Disc. Math. 59:135–166.

    Article  Google Scholar 

  • Shimbel, A. (1954). Structure in communication nets. Proc. Symp. on Information Networks. Polytechnic Inst. of Brooklyn, 119–203.

    Google Scholar 

  • Sridharan, S. (1993). On the strong path partition conjecture of Berge. Discrete Mathematics 117:265–270.

    Article  Google Scholar 

  • Szekeres, C, and Wilf, H.S. (1968). An inequality for the chromatic number of a graph. Journal of Combinatorial Theory 4:1–3.

    Article  Google Scholar 

  • Tabourier, Y. (1973). All shortest distances in a graph. An improvement to Dantzig’s inductive algorithm. Discrete Mathematics 4:83–87.

    Article  Google Scholar 

  • Takaoka, T. (1998). Subcubic cost algorithms for the all pairs shortest path problem. Algorithmica 20:309–318.

    Article  Google Scholar 

  • Thomassen, C. (1980). Hamiltonian-connected tournaments. J. Combin. Theory B28:142–163.

    Google Scholar 

  • Tomescu, I. (1968). Sur l’algorithme matriciel de B. Roy. Revue Française Informat. Rech. opér. 2:87–91.

    Google Scholar 

  • Warren, H.S. (1975). A modification of Warshall’s algorithm for the transitive closure of binary relations. Comm. ACM 18:218–220.

    Article  Google Scholar 

  • Warshall, S. (1962). A theorem on Boolean matrices. J. of ACM 9:11–12.

    Article  Google Scholar 

  • Wongseelashote, A. (1976). An algebra for determining all path-values in a network with applications to Kshortest paths problem. Networks 6:307–334.

    Article  Google Scholar 

  • Yoeli, M. (1961). A note on a generalization of Boolean matrix theory. American Math. Monthly 68:552–557.

    Article  Google Scholar 

  • Yuval, G. (1975/76). An algorithm for finding all shortest paths using N 2.81 infinite-precision multiplications. Information Processing Lett. 4:155–156.

    Article  Google Scholar 

  • Zhang, K.M., and Song, Z.M. (1991). Cycles in digraphs — A survey. Nanjing Daxue Xuebao Ziran Kexue Ban, (Special Issue) 27:188–215.

    Google Scholar 

  • Zhu, S.Y. (1985). A parallel computation of the transitive closure of a relation using Warshall’s method. J. Shanghai Jiaotong Univ. 19:101–107, 127.

    Google Scholar 

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Denis Bouyssou Eric Jacquet-Lagrèze Patrice Perny Roman Słowiński Daniel Vanderpooten Philippe Vincke

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Hansen, P., de Werra, D. (2002). Connectivity, Transitivity and Chromaticity: The Pioneering Work of Bernard Roy in Graph Theory. In: Bouyssou, D., Jacquet-Lagrèze, E., Perny, P., Słowiński, R., Vanderpooten, D., Vincke, P. (eds) Aiding Decisions with Multiple Criteria. International Series in Operations Research & Management Science, vol 44. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0843-4_2

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  • DOI: https://doi.org/10.1007/978-1-4615-0843-4_2

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