Missouri Journal of Mathematical Sciences

An Annotated Bibliography on the Thickness, Outerthickness, and Arboricity of a Graph

Erkki Mäkinen and Timo Poranen

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Abstract

The bibliography introduces literature on graph thickness, outerthickness, and arboricity. In addition to the pointers to the literature we also give some conjectures concerning known open problems on the field.

Article information

Source
Missouri J. Math. Sci. Volume 24, Issue 1 (2012), 76-87.

Dates
First available in Project Euclid: 25 May 2012

Permanent link to this document
http://projecteuclid.org/euclid.mjms/1337950501

Mathematical Reviews number (MathSciNet)
MR2977132

Subjects
Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

Keywords
Thickness outerthickness arboricity

Citation

Mäkinen, Erkki; Poranen, Timo. An Annotated Bibliography on the Thickness, Outerthickness, and Arboricity of a Graph. Missouri J. Math. Sci. 24 (2012), no. 1, 76--87. http://projecteuclid.org/euclid.mjms/1337950501.


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References

  • A. Aggarwal, M. Klawe, and P. Shor, Multilayer grid embeddings for VLSI, Algorithmica, 6.1 (1991), 129–151.
  • I. Aho, E. Mäkinen, and T. Systä, Remarks on the thickness of a graph, Information Sciences, 108 (1998), 1–4.
  • M. O. Albertson, D. Boutin, and E. Gethner, The thickness and chromatic number of r-inflated graphs, Discrete Mathematics, 310.20 (2010), 2725–2734.
  • V. B. Alekseev amd V. S. Gonchakov, Thickness for arbitrary complete graphs, Matematicheskij Sbornik, 143 (1976), 212–230.
  • K. Asano, On the genus and thickness of graphs, Journal of Combinatorial Theory Series B, 43 (1987), 287–292.
  • K. Asano, On the thickness of graphs with genus 2, Ars Combinatorica, 38 (1994), 87–95.
  • J. Battle, F. Harary, and Y. Kodoma, Every planar graph with nine points has a non-planar complement, Bulletin of the American Mathematical Society, 68 (1962), 569–571.
  • L. W. Beineke, Minimal decompositions of complete graphs into subgraphs with embeddability properties, Canadian Journal of Mathematics, 21 (1969), 992–1000.
  • L. W. Beineke, Complete bipartite graphs: decomposition into planar subgraphs, in F. Haray and L. W. Beineke, editors, A Seminar on Graph Theory, Holt, Rinehar and Winston, 1970, pp. 42–53.
  • L. W. Beineke, Fruited planes, Congressus Numerantium, 63 (1988), 127–138.
  • L. W. Beineke, Biplanar graphs: a survey, Computers & Mathematics with Applications, 34.11 (1997), 1–8.
  • L. W. Beineke and F. Harary, Inequalities involving the genus of a graph and its thicknesses, Proceedings of the Glasgow Mathematical Association, 7 (1965), 19–21.
  • L. W. Beineke and F. Harary, The thickness of the complete graph, Canadian Journal of Mathematics, 17 (1965), 850–859.
  • L. W. Beineke, F. Harary, and J. W. Moon, On the thickness of the complete bipartite graphs, Proceedings of the Cambridge Philosophical Society, 60 (1964), 1–5.
  • P. Bose and K. A. Prabhu, Thickness of graphs with degree constrained vertices, IEEE Transactions on Circuits and Systems, 24.4 (1977), 184–190.
  • C. Bourke, R. Tewari, and N. V. Vinodchandran, Directed planar reachability is in unambiguous log-space, ACM Transactions on Computation Theory, 1.1 (2009), Article No. 4.
  • D. L. Boutin, E. Gethner, and T. Sulanke, Thickness-two graphs part one: new nine-critical graphs, permuted layer graphs, and Catlin's graphs, Journal of Graph Theory, 57.3 (2008), 198–214.
  • G. Chartrand, D. Geller, and S. Hedetniemi, Graphs with forbidden subgraphs, Journal of Combinatorial Theory, 10B (1971), 12–41.
  • R. Cimikowski, On heuristics for determining the thickness of a graph, Information Sciences, 85 (1995), 87–98.
  • C. Cooper, On the thickness of sparse random graphs, Combinatorics Probability and Computing, 1 (1992), 303–309.
  • É. Czabarka, O. Sýkora, L. A. Székely, and I. Vrt'o, Biplanar crossing numbers II, Comparing crossing numbers and biplanar crossing numbers using the probabilistic method, Random Structures and Algorithms, 33 (2008), 480–496.
  • A. M. Dean, W. Evans, E. Gethner, J. D. Laison, M. A. Safari, and W. Trotter, Bar k-visibility graphs: Bounds on the number of edges, chromatic number, and thickness, Journal of Graph Algorithms and Applications, 11.1 (2007), 45–59.
  • A. M. Dean and J. P. Hutchinson, On some variations of the thickness of a graph connected with colouring, in Proceedings of International Conference on the Theory and Applications of Graphs, Vol. 6, (1991), pp. 287–296.
  • A. M. Dean, J. P. Hutchinson, and E. R. Scheinerman, On the thickness and arboricity of a graph, Journal of Combinatorial Theory Series B, 52 (1991), 147–151.
  • V. Dujmović and D. R. Wood, Graph treewidth and geometric thickness parameters, Discrete and Computational Geometry, 37 (2007), 641–670.
  • D. Eppstein, Separating thickness from geometric thickness, in Proceedings of the 10th International Symposium on Graph Drawing, Volume 2528 of Lecture Notes in Computer Science, (2002), pp. 150–162.
  • D. Eppstein, Testing bipartiteness of geometric intersection graphs, ACM Transactions on Algorithms, 5.2 (2009), Article No. 15.
  • E. Gethner and T. Sulanke, Thickness-two graphs part two: more new nine-critical graphs, independence ratio, cloned planar graphs, and singly and doubly outerplanar graphs, Graphs and Combinatorics, 25 (2009), 197–217.
  • J. H. Halton, On the thickness of graphs of given degree, Information Sciences, 54 (1991), 219–238.
  • F. Harary, A research problem, Bulletin of the American Mathematical Society, 67 (1961), 542.
  • F. Harary, Graph Theory, Addison-Wesley, 1971.
  • A. M. Hobbs, A survey of thickness, in Recent Progress in Combinatorics (Proceedings of the 3rd Waterloo Conference on Combinatorics, 1968, 1969, pp. 255–264.
  • P. Horák and J. Širá\un, On a modified concept of a thickness of a graph, Mathematische Nachrichten, 108 (1982), 305–306.
  • J. P. Hutchinson, T. Shermer, and A. Vince, On representations of some thickness-two graphs, Computational Geometry, 131 (1999), 161–171.
  • M. Jünger, P. Mutzel, T. Odenthal, and M. Scharbrodt, The thickness of minor-excluded class of graphs, Discrete Mathematics, 182 (1998), 169–176.
  • P. C. Kainen, Thickness and coarseness of graphs, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 39 (1973), 88–95.
  • A. Kaveh and H. Rahami, An efficient algorithm for embedding nonplanar graphs in planes, Journal of Mathematical Modelling and Algorithms, 1 (2002), 257–268.
  • S. Kawano and K. Yamazaki, Worst case analysis of a greedy algorithm for the graph thickness, Information Processing Letters, 85 (2003), 333–337.
  • M. Kleinert, Die Dicke des $n$-dimensionale Würfel-Graphen, Journal of Combinatorial Theory, 3 (1967), 10–15.
  • A. Kotzig, On certain decompositions of graphs, Matematicko-Fyzikálny \uCasopis, 5 (1955), 144–151.
  • A. Liebers, Planarizing graphs - a survey and annotated bibliography, Journal of Graph Algorithms and Applications, 5 (2001), 1–74.
  • E. Mäkinen, T. Poranen, and P. Vuorenmaa, A genetic algorithm for determining the thickness of a graph, Information Sciences, 138 (2001), 155–164.
  • A. Mansfield, Determining the thickness of graphs is NP-hard, Mathematical Proceedings of the Cambridge Philosophical Society, 93 (1983), 9–23.
  • M. Massow and S. Felsner, Thickness of bar 1-visibility graphs, in Proceedings of the 15th International Symposium on Graph Drawing, Volume 2528 of Lecture Notes in Computer Science, 2007, pp. 330–342.
  • J. Mayer, Décomposition de $K_{16}$ en Trois Graphes Planaires, Journal of Combinatorial Theory Series B, 13 (1972), 71.
  • P. Mutzel, T. Odenthal, and M. Scharbrodt, The thickness of graphs: a survey, Graphs and Combinatorics, 14 (1998), 59–73.
  • T. Poranen, Approximation Algorithms for Some Topological Invariants of Graphs, Ph.D. thesis, University of Tampere, 2004.
  • T. Poranen, A simulated annealing algorithm for determining the thickness of a graph, Information Sciences, 172 (2005), 155–172.
  • T. Poranen, Two new approximation algorithms for the maximum planar subgraph problem, Acta Cybernetica, 18.3 (2008), 503–527.
  • T. Poranen and E. Mäkinen, Remarks on the thickness and outerthickness of a graph, Computers & Mathematics with Applications, 50 (2005), 249–254.
  • S. Ramanathan and E. L. Lloyd, Scheduling algorithms for multihop radio networks, IEEE/ACM Transactions on Networking, 1 (1993), 166–177.
  • G. Ringel, Die torodiale Dicke des vollständigen Graphen, Mathematische Zeitschrift, 87 (1965), 19–26.
  • J. Širá\un and P. J. Horák, A construction of thickness-minimal graphs, Discrete Mathematics, 64 (1987), 262–268.
  • O. Sýkora, L. A. Székely, and I. Vrt'o, A note on Halton's conjecture, Information Sciences, 164.1–4 (2004), 61–64.
  • W. T. Tutte, The non-biplanar character of the complete 9-graph, Canadian Mathematical Bulletin, 6 (1963), 319–330.
  • W. T. Tutte, The thickness of a graph, Indagationes Mathematicae, 25 (1963), 567–577.
  • J. M. Vasak, The thickness of the complete graph, Notices of the American Mathematical Society, 23 (1976), A-479, Abstract.
  • W. Wessel, On some variations of the thickness of a graph connected with colouring, in Graphs and Other Combinatorial Topics. Proceeding of the Third Czechoslovak Symposium on Graph Theory, Volume 59 of Teubner, Texte zur Mathematik, 1983, p. 344–348.
  • W. Wessel, Über die Abhängigkeit der Dicke eines Graphen von seinen Knotenpunktvalenzen, Geometrie und Kombinatorik, 2.2 (1984), 235–238.
  • A. T. White and L. W. Beineke, Topological graph theory, in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, 1978, pp. 15–49.
  • D. Gonçalves, Edge partition of planar graphs into two outerplanar graphs, in Proceedings of the Annual ACM Symposium on Theory of Computing, 2005, pp. 504–512.
  • R. Guy, Outerthickness and outercoarseness of graphs, in Proc. British Combinatorial Conference, Volume 13 of London Mathematics Society Lecture Note Series, 1974, pp. 57–60.
  • R. K. Guy and R. J. Nowakowski, The outerthickness and outercoarseness of graphs I. The complete graph & the $n$-cube, in R. Bodendiek and R. Henns, editors, Topics of Combinatorics and Graph Theory: Essays in Honour of Gerhard Ringel, Physica-Verlag, 1990, pp. 297–310.
  • R. K. Guy and R. J. Nowakowski, The outerthickness and outercoarseness of graphs II. The complete bipartite graph, in R. Bodendiek, editor, Contemporary Methods in Graph Theory, B. I. Wissenchaftsverlag, 1990, pp. 313–322.
  • L. S. Heath, Edge coloring planar graphs with two outerplanar subgraphs, in Proceedings of the 2nd ACM-SIAM Symposium on Discrete Algorithms, 1991, pp. 195–202.
  • K. S. Kedlaya, Outerplanar partitions of planar graphs, Journal of Combinatorial Theory, Series B, 67.2 (1996), 238–248.
  • J. Akiyama, G. Exoo, and F. Harary, Covering and packing in graphs IV: Linear arboricity, Networks, 11 (1981), 69–72.
  • J. Akiyama and T. Hamada, The decompositions of line graphs, middle graphs and total graphs of complete graphs into forests, Discrete Mathematics, 26 (1979), 203–208.
  • N. Alon, C. McDiarmid, and B. Reed, Star arboricity, Combinatorica, 12 (1992), 375–380.
  • T. Bartnicki, J. Grytczuk, and H. Kierstead, The game of arboricity, Discrete Mathematics, 308 (2008), 1388–1393.
  • L. W. Beineke, Decompositions of complete graphs into forests, Magyar Tud. Akad. Mat. Kutató Int. Közl., 9 (1965), 589–594.
  • T. Biedl and F. Brandenburg, Partitions of graphs into trees, in Proc. of the 10th International Symposium on Graph Drawing, Lecture Notes in Computer Science, 2006, pp. 430–439.
  • P. Bose, F. Hurtado, E. Rivera-Campo, and D. R. Wood, Partitions of complete geometric graphs into plane trees, Computational Geometry: Theory and Applications, 34 (2006), 116–125.
  • G. J. Chang, C. Chen, and Y. Chen, Vertex and tree arboricities of graphs, Journal of Combinatorial Optimization, 8 (2004), 295–306.
  • B. Chen, M. Matsumoto, J. Wang, Z. Zhang, and J. Zhang, A short proof of Nash-Williams' theorem for the arboricity of a graph, Graphs and Combinatorics, 10 (1994), 27–28.
  • M. Cygan, L. Kowalik, and B. L\uuzar, A planar linear arboricity conjecture, 2009, Preprint at \tthttp://arxiv.org/pdf/0912.5528.
  • E. S. El-Mallah and C. J. Colbourn, Partitioning the edges of a planar graph into two partial $k$-trees, Congressus Numerantium, 66 (1988), 69–80.
  • H. L. Fu, K.-C. Huang, and C-H. Yen, The linear 3-arboricity of $K_{n,n}$ and $K_n$, Discrete Mathematics, 308 (2007), 3816–3823.
  • A. Garcí\ia, C. Hernando, M. Hurtado, F. Noy, and J. Tejel, Packing trees into planar graphs, Journal of Graph Theory, 40 (2002), 172–181.
  • D. Gonçalves, Caterpillar arboricity of planar graphs, Discrete Mathematics, 307 (2007), 2112–2121.
  • D. Gonçalves, Covering planar graphs with forests, one having bounded maximum degree, Journal of Combinatorial Theory, Series B, 99 (2008), 314–322.
  • D. Gonçalves and P. Ochem, On some arboricities in planar graphs, Electronic Notes in Discrete Mathematics, 22 (2005), 427–432.
  • D. Gonçalves and P. Ochem, On star and catepillar arboricities, Discrete Mathematics, 309 (2009), 3694–4702.
  • D. Gonçalves, A. Pinlou, and S. Thomassé, Spanning galaxies in digraphs, Electronic Notes in Discrete Mathematics, 34 (2009), 139–143.
  • R. Haas, Characterizations of arboricity of graphs, Ars Combinatoria, 63 (2002), 129–138.
  • Q. Liu and D. B. West, Tree-thickness and caterpillar-thickness under girth constraints, The Electronic Journal of Combinatorics, 15 (2008), 1–11.
  • M. Montassier, P. O. de Mendez, A. Raspaud, and X. Zhu, Decomposing a graph into forests, Journal of Combinatorial Theory, Series B, 102 (2012), 38–52.
  • C. St. J. Nash-Williams, Edge-disjoint spanning trees of finite graphs, Journal of London Mathematical Society, 36 (1961), 445–450.
  • C. St. J. Nash-Williams, Decomposition of finite graphs into forests, Journal of London Mathematical Society, 39 (1964), 12.
  • V. Petrovic, Decomposition of some planar graphs into trees, in Proceedings of International Conference on Combinatorics, 1993, p. 48.
  • G. Ringle, Two trees in maximal planar bipartite graphs, Journal of Graph Theory, 17 (1993), 755–758.
  • M. J. Stein, Arboricity and tree-packing in locally finite graphs, Journal of Combinatorial Theory, Series B, 96 (2006), 302–312.
  • I. Streinu and L. Theran, Sparsity-certifying graph decompositions, Graphs and Combinatorics, 25 (2009), 219–238.
  • J.-L. Wu and Y.-W. Wu, The linear arboricity of planar graphs of maximum degree seven is four, Journal of Graph Theory, 58 (2008), 201–210.
  • D. Yang and H. A. Kierstead, Asymmetric marking games on line graphs, Discrete Mathematics, 308 (2008), 1751–1755.