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Further Reading
M.C. Golumbic, “Algorithmic Graph Theory and Perfect Graphs”, Academic Press, New York, 1980. Second edition: Annals of Discrete Mathematics 57, Elsevier, Amsterdam, 2004. This has now become the classic introduction to the field. It conveys the message that intersection graph models are a necessary and important tool for solving real-world problems for a large variety of application areas, and on the mathematical side, it has provided rich soil for deep theoretical results in graph theory. In short, it remains a stepping stone from which the reader may embark on one of many fascinating research trails. The second edition of Algorithmic Graph Theory and Perfect Graphs includes a new chapter called Epilogue 2004 which surveys much of the new research directions from the Second Generation. Its intention is to whet the appetite. Seven other books stand out as the most important works covering advanced research in this area. They are the following, and are a must for any graph theory library.
A. Brandstädt, V.B. Le and J.P. Spinrad, “Graph Classes: A Survey”, SIAM, Philadelphia, 1999, This is an extensive and invaluable compendium of the current status of complexity and mathematical results on hundreds of families of graphs. It is comprehensive with respect to definitions and theorems, citing over 1100 references.
P.C. Fishburn, “Interval Orders and Interval Graphs: A Study of Partially Ordered Sets”, John Wiley & Sons, New York, 1985. Gives a comprehensive look at the research on this class of ordered sets.
M.C. Golumbic and A.N. Trenk, “Tolerance Graphs”, Cambridge University Press, 2004. This is the youngest addition to the perfect graph bookshelf. It contains the first thorough study of tolerance graphs and tolerance orders, and includes proofs of the major results which have not appeared before in books.
N.V.R. Mahadev and U.N. Peled, “Threshold Graphs and Related Topics”, North-Holland, 1995. A thorough and extensive treatment of all research done in the past years on threshold graphs (Chapter 10 of Golumbic [1]), threshold dimension and orders, and a dozen new concepts which have emerged.
T.A. McKee and F.R. McMorris, “Topics in Intersection Graph Theory”, SIAM, Philadelphia, 1999. A focused monograph on structural properties, presenting definitions, major theorems with proofs and many applications.
F.S. Roberts, “Discrete Mathematical Models, with Applications to Social, Biological, and Environmental Problems”, Prentice-Hall, Engelwood Cliffs, New Jersey, 1976. This is the classic book on many applications of intersection graphs and other discrete models.
W.T. Trotter, “Combinatorics and Partially Ordered Sets”, Johns Hopkins, Baltimore, 1992. Covers new directions of investigation and goes far beyond just dimension problems on ordered sets.
Additional References
E. Bibelnieks and P.M. Dearing, Neighborhood subtree tolerance graphs, Discrete Applied Math. 43: 13–26 (1993).
P.C. Gilmore and A.J. Hoffman, A characterization of comparability graphs and of interval graphs. Canad. J. Math. 16: 539–548 (1964).
M.C. Golumbic and R.E. Jamison, The edge intersection graphs of paths in a tree, J. Combinatorial Theory, Series B 38: 8–22 (1985).
M.C. Golumbic and R.E. Jamison, Edge and vertex intersection of paths in a tree, Discrete Math. 55: 151–159 (1985).
M.C. Golumbic and H. Kaplan and R. Shamir, Graph sandwich problems, J. of Algorithms 19: 449–473 (1995).
M.C. Golumbic and R.C. Laskar, Irredundancy in circular arc graphs, Discrete Applied Math. 44: 79–89 (1993).
M.C. Golumbic, M. Lipshteyn and M. Stern, The k-edge intersection graphs of paths in a tree, Congressus Numerantium (2004), to appear.
M.C. Golumbic and C.L. Monma, A generalization of interval graphs with tolerances”, Congressus Numerantium 35: 321–331 (1982).
M.C. Golumbic and C.L. Monma and W.T. Trotter, Tolerance graphs, Discrete Applied Math. 9: 157–170 (1984).
M.C. Golumbic and R. Shamir, Complexity and algorithms for reasoning about time: A graph theoretic approach, J. Assoc. Comput. Mach. 40: 1108–1133 (1993).
M.C. Golumbic and A. Siani, Coloring algorithms and tolerance graphs: reasoning and scheduling with interval constraints, Lecture Notes in Computer Science 2385, Springer-Verlag, (2002) pp. 196–207.
R.E. Jamison and H.M. Mulder, Constant tolerance representations of graphs in trees, Congressus Numerantium 143: 175–192 (2000).
V. Klee, What are the intersection graphs of arcs in a circle? American Math. Monthly 76: 810–813 (1969).
L. Langley, Interval tolerance orders and dimension, Ph.D. Thesis, Dartmouth College”, June, (1993).
F.R. McMorris and D.R. Shier, Representing chordal graphs on K1,n, Comment. Math. Univ. Carolin. 24: 489–494 (1983).
G. Narasimhan and R. Manber, Stability number and chromatic number of tolerance graphs, Discrete Applied Math. 36: 47–56 (1992).
R.E. Tarjan, Decomposition by clique separators, Discrete Math. 55: 221–232 (1985).
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Golumbic, M.C. (2005). Algorithmic Graph Theory and Its Applications. In: Golumbic, M.C., Hartman, I.BA. (eds) Graph Theory, Combinatorics and Algorithms. Operations Research/Computer Science Interfaces Series, vol 34. Springer, Boston, MA. https://doi.org/10.1007/0-387-25036-0_3
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