New Directions in Graph Theory (With an Emphasis on the Role of Applications)

Abstract We summarize some general themes which we saw as coming out of Quo Vadis Graph Theory? — an international conference on the future of graph theory, held at the University of Alaska, Fairbanks, in August 1990. We expand on these themes with specific examples and emphasize the role of applications.

[1]  Edward R. Scheinerman,et al.  Random interval graphs , 1988, Comb..

[2]  J. Aczél,et al.  On scientific laws without dimensional constants , 1986 .

[3]  S. Stahl.  n-Tuple colorings and associated graphs , 1976 .

[4]  Peter C. Fishburn.  On the sphericity and cubicity of graphs , 1983, J. Comb. Theory, Ser. B.

[5]  Ilia Krasikov,et al.  The reconstruction of a tree from its number deck , 1985, Discret. Math..

[6]  Fred S. Roberts,et al.  On the Optimal Strongly Connected Orientations of City Street Graphs I: Large Grids , 1988, SIAM J. Discret. Math..

[7]  Al Young.  Providence, Rhode Island , 1975 .

[8]  Jørgen Bang-Jensen,et al.  The effect of two cycles on the complexity of colourings by directed graphs , 1989, Discret. Appl. Math..

[9]  Arundhati Raychaudhuri.  Optimal Multiple Interval Assignments in Frequency Assignment and Traffic Phasing , 1992, Discret. Appl. Math..

[10]  András Gyárfás,et al.  On-line and first fit colorings of graphs , 1988, J. Graph Theory.

[11]  Victor Klee,et al.  Sign-Patterns and Stability , 1989 .

[12]  Edward R. Scheinerman,et al.  An evolution of interval graphs , 1990, Discret. Math..

[13]  Mikhail J. Atallah,et al.  Parallel Strong Orientation of an Undirected Graph , 1984, Inf. Process. Lett..

[14]  Ron Shamir,et al.  Probabilistic Analysis in Linear Programming , 1993 .

[15]  Michael Zuker,et al.  The sensitivity of eigenvalues under elementary matrix perturbations , 1987 .

[16]  David S. Johnson,et al.  The Complexity of Computing Steiner Minimal Trees , 1977 .

[17]  W. H. Day,et al.  Critical comparison of consensus methods for molecular sequences. , 1992, Nucleic acids research.

[18]  Jaroslav Nesetril,et al.  On the complexity of H-coloring , 1990, J. Comb. Theory, Ser. B.

[19]  K. Borgwardt.  A probabilistic analysis of the simplex method , 1986 .

[20]  László Lovász,et al.  Kneser's Conjecture, Chromatic Number, and Homotopy , 1978, J. Comb. Theory A.

[21]  M. Trick,et al.  Voting schemes for which it can be difficult to tell who won the election , 1989 .

[22]  Eric S. Lander,et al.  The distribution of clusters in random graphs , 1990 .

[23]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[24]  A. Tucker,et al.  Perfect Graphs and an Application to Optimizing Municipal Services , 1973 .

[25]  Alexander Schrijver,et al.  Matrix Cones, Projection Representations, and Stable Set Polyhedra , 1990, Polyhedral Combinatorics.

[26]  Lawrence Bodin,et al.  Networks and vehicle routing for municipal waste collection , 1974, Networks.

[27]  Alexander Schrijver.  Tait's Flyping Conjecture for Well-Connected Links , 1993, J. Comb. Theory, Ser. B.

[28]  Fred S. Roberts,et al.  Meaningless Statements, Matching Experiments, and Colored Digraphs , 1989 .

[29]  Edward R. Scheinerman,et al.  Random intervals , 1990 .

[30]  P. Erdös,et al.  On the structure of linear graphs , 1946 .

[31]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[32]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[33]  S. Benzer.  The fine structure of the gene. , 1962, Scientific American.

[34]  Eugene C. Johnsen,et al.  The Micro-Macro Connection: Exact Structure and Process , 1989 .

[35]  W. K. Hale.  Frequency assignment: Theory and applications , 1980, Proceedings of the IEEE.

[36]  R. Shamir.  The Efficiency of the Simplex Method: A Survey , 1987 .

[37]  Fred S. Roberts,et al.  Building and Analyzing an Energy Demand Signed Digraph , 1973 .

[38]  László Lovász,et al.  Normal hypergraphs and the perfect graph conjecture , 1972, Discret. Math..

[39]  Fred S. Roberts,et al.  Structural modeling and measurement theory , 1979 .

[40]  Maolin Zheng,et al.  A revised peeling algorithm for determining if a hexagonal system is Kekuléan , 1991 .

[41]  Harvey J. Greenberg,et al.  Inverting Signed Graphs , 1984 .

[42]  Fred S. Roberts,et al.  I-Colorings, I-Phasings, and I-Intersection assignments for graphs, and their applications , 1983, Networks.

[43]  Fred S. Roberts,et al.  On the Optimal Strongly Connected Orientations of City Street Graphs IV: Four East-West Avenues or North-South Streets , 1994, Discret. Appl. Math..

[44]  F. K. Hwang,et al.  Most reliable double loop networks in survival reliability , 1993, Networks.

[45]  Andrzej Rucinski,et al.  Ramsey properties of random graphs , 1992, J. Comb. Theory, Ser. B.

[46]  Svante Janson,et al.  Small Cliques in Random Graphs , 1990, Random Struct. Algorithms.

[47]  Fanica Gavril,et al.  Constructing Trees with Prescribed Cardinalities for the Components of Their Vertex Deleted Subgraphs , 1985, J. Algorithms.

[48]  B. Bollobás.  The evolution of random graphs , 1984 .

[49]  Frank Harary,et al.  On double and multiple interval graphs , 1979, J. Graph Theory.

[50]  Fred S. Roberts,et al.  Optimal I-Intersection assignments for graphs: A linear programming approach , 1983, Networks.

[51]  Fred S. Roberts,et al.  On dimensional properties of graphs , 1989, Graphs Comb..

[52]  Fred S. Roberts,et al.  Weighted Digraph Models for the Assessment of Energy Use and Air Pollution in Transportation Systems , 1975 .

[53]  J. Michael Steele,et al.  Probabilistic and Worst Case Analyses of Classical Problems of Combinatorial Optimization in Euclidean Space , 1990, Math. Oper. Res..

[54]  Peter L. Hammer,et al.  Difference graphs , 1990, Discret. Appl. Math..

[55]  S. J. Cyvin,et al.  Kekule Structures in Benzenoid Hydrocarbons , 1988 .

[56]  Elzbieta Bozena Jarrett.  Transformations of graphs and digraphs , 1991 .

[57]  Timothy F. Havel,et al.  The combinatorial distance geometry method for the calculation of molecular conformation. I. A new approach to an old problem. , 1983, Journal of theoretical biology.

[58]  Zoltán Füredi,et al.  Extremal problems concerning Kneser graphs , 1986, J. Comb. Theory, Ser. B.

[59]  Pierre Hansen,et al.  Algorithms for Voting and Competitive Location on a Network , 1988, Transp. Sci..

[60]  L. Freeman.  Centrality in social networks conceptual clarification , 1978 .

[61]  Louis V. Quintas,et al.  The number of alkanes having n carbons and a longest chain of length d: An application of a theorem of Polya , 1988 .

[62]  Eli Upfal,et al.  Constructing a perfect matching is in random NC , 1985, STOC '85.

[63]  Pavol Hell,et al.  On multiplicative graphs and the product conjecture , 1988, Comb..

[64]  J. Aczél,et al.  On the possible merging functions , 1989 .

[65]  Uzi Vishkin,et al.  On Efficient Parallel Strong Orientation , 1985, Inf. Process. Lett..

[66]  Ron Holzman,et al.  An Axiomatic Approach to Location on Networks , 1990, Math. Oper. Res..

[67]  Timothy F. Havel,et al.  The combinatorial distance geometry method for the calculation of molecular conformation. II. Sample problems and computational statistics. , 1983, Journal of theoretical biology.

[68]  Joseph E. McCanna.  Orientations of the n-cube with minimum diameter , 1988, Discret. Math..

[69]  Louis Narens,et al.  Meaningfulness and the Erlanger program of Felix Klein , 1988 .

[70]  J. Richard Lundgren,et al.  Food Webs, Competition Graphs, Competition-Common Enemy Graphs, and Niche Graphs , 1989 .

[71]  Fred S. Roberts,et al.  Applications of the theory of meaningfulness to psychology , 1985 .

[72]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[73]  K. Bogart.  Preference structures I: Distances between transitive preference relations† , 1973 .

[74]  Fred S. Roberts,et al.  Signed Digraphs and the Growing Demand for Energy , 1971 .

[75]  L. Lovász.  A Characterization of Perfect Graphs , 1972 .

[76]  D. A. Neumann,et al.  Consensus functions defined on trees , 1983 .

[77]  Fred S. Roberts,et al.  STRUCTURE AND STABILITY IN WEIGHTED DIGRAPH MODELS * , 1979 .

[78]  Fred S. Roberts,et al.  T-colorings of graphs: recent results and open problems , 1991, Discret. Math..

[79]  F. Boesch,et al.  ROBBINS'S THEOREM FOR MIXED MULTIGRAPHS , 1980 .

[80]  B. Rothschild,et al.  Every graph is contained in a sparsest possible balanced graph , 1985, Mathematical Proceedings of the Cambridge Philosophical Society.

[81]  V. Klee,et al.  When is a Matrix Sign Stable? , 1977, Canadian Journal of Mathematics.

[82]  P. Arabie,et al.  An algorithm for clustering relational data with applications to social network analysis and comparison with multidimensional scaling , 1975 .

[83]  Harvey J. Greenberg,et al.  Computer-assisted analysis and model simplification , 1981 .

[84]  Lowell W. Beineke,et al.  A classification scheme for vulnerability and reliability parameters of graphs , 1993 .

[85]  John S. Maybee,et al.  Qualitatively Stable Matrices and Convergent Matrices , 1989 .

[86]  J. T. Madison,et al.  Structure of a Ribonucleic Acid , 1965, Science.

[87]  Fred S. Roberts,et al.  On the optimal strongly connected orientations of city street graphs. III. Three east-west avenues or north-south streets , 1989, Networks.

[88]  Martin E. Dyer,et al.  Randomized Greedy Matching , 1991, Random Struct. Algorithms.

[89]  John Paul Boyd,et al.  Social semigroups : a unified theory of scaling and blockmodelling as applied to social networks , 1992 .

[90]  B. Jaumard,et al.  Minimum sum of diameters clustering , 1987 .

[91]  Richard M. Karp,et al.  Parallel Algorithms for Shared-Memory Machines , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[92]  Fred S. Roberts,et al.  Meaningfulness of conclusions from combinatorial optimization , 1990, Discret. Appl. Math..

[93]  William H. E. Day,et al.  A formalization of consensus index methods , 1985 .

[94]  Kenneth P. Bogart,et al.  Consensus Signed Digraphs , 1979 .

[95]  F. McMorris,et al.  The median procedure for n-trees , 1986 .

[96]  Danail Bonchev,et al.  The graph center concept for polycyclic graphs , 1981 .

[97]  B. Bollobás.  Threshold functions for small subgraphs , 1981 .

[98]  David Kelly,et al.  Graph coloring satisfying restraints , 1990, Discret. Math..

[99]  Hiroshi Maehara,et al.  A digraph represented by a family of boxes or spheres , 1984, J. Graph Theory.

[100]  F. Harary.  On the notion of balance of a signed graph. , 1953 .

[101]  B. Jaumard,et al.  Maximum sum-of-splits clustering , 1989 .

[102]  Phipps Arabie,et al.  Constructing blockmodels: How and why , 1978 .

[103]  Fred R. McMorris,et al.  Consensusn-trees , 1981 .

[104]  Jan Karel Lenstra,et al.  Probabilistic analysis of combinatorial algorithms: an annotated bibliography , 1984 .

[105]  C DeLisi,et al.  Computers in molecular biology: current applications and emerging trends. , 1988, Science.

[106]  Béla Bollobás,et al.  List-colourings of graphs , 1985, Graphs Comb..

[107]  Alexander Schrijver,et al.  Relaxations of vertex packing , 1986, J. Comb. Theory B.

[108]  Hiroshi Maehara.  Sphericity exceeds cubicity for almost all complete bipartite graphs , 1986, J. Comb. Theory, Ser. B.

[109]  F. Roberts.  Graph Theory and Its Applications to Problems of Society , 1987 .

[110]  G. Hutchinson,et al.  Evaluation of polymer sequence fragment data using graph theory. , 1969, The Bulletin of mathematical biophysics.

[111]  Carsten Thomassen,et al.  Distances in orientations of graphs , 1975, J. Comb. Theory, Ser. B.

[112]  S. Benzer.  ON THE TOPOLOGY OF THE GENETIC FINE STRUCTURE. , 1959, Proceedings of the National Academy of Sciences of the United States of America.

[113]  Hiroshi Maehara,et al.  Space graphs and sphericity , 1984, Discret. Appl. Math..

[114]  Pak-Ken Wong,et al.  Cages - a survey , 1982, J. Graph Theory.

[115]  Pierre Hansen,et al.  Unimodular functions , 1986, Discrete Applied Mathematics.

[116]  W. Thurston.  The geometry and topology of 3-manifolds , 1979 .

[117]  B. Jaumard,et al.  Efficient algorithms for divisive hierarchical clustering with the diameter criterion , 1990 .

[118]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[119]  Robert E. Tarjan,et al.  Strongly connected orientations of mixed multigraphs , 1985, Networks.

[120]  K. Bogart.  Preference Structures. II: Distances Between Asymmetric Relations , 1975 .

[121]  S. Borgatti,et al.  LS sets, lambda sets and other cohesive subsets , 1990 .

[122]  Alan M. Frieze.  On the Exact Solution of Random Travelling Salesman Problems with Medium Size Integer Coefficients , 1987, SIAM J. Comput..

[123]  F. Harary,et al.  STRUCTURAL BALANCE: A GENERALIZATION OF HEIDER'S THEORY1 , 1977 .

[124]  Oded Schramm,et al.  Existence and uniqueness of packings with specified combinatorics , 1991 .