An annotated bibliography on the thickness, outerthickness, and arboricity of a graph

This 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.

[1]  C. Nash-Williams.  Edge-disjoint spanning trees of finite graphs , 1961 .

[2]  F. Harary,et al.  Every planar graph with nine points has a nonplanar complement , 1962 .

[3]  W. T. Tutte.  The Non-Biplanar Character of the Complete 9-Graph , 1963, Canadian Mathematical Bulletin.

[4]  W. T. Tutte,et al.  The thickness of a graph , 1963 .

[5]  C. Nash-Williams.  Decomposition of Finite Graphs Into Forests , 1964 .

[6]  J. Moon,et al.  On the thickness of the complete bipartite graph , 1964, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  L. Beineke,et al.  On the thickness of the complete graph , 1964 .

[8]  L. Beineke,et al.  Inequalities involving the genus of a graph and its thicknesses , 1965, Proceedings of the Glasgow Mathematical Association.

[9]  G. Ringel,et al.  Die toroidale Dicke des vollständigen Graphen , 1965 .

[10]  M. Kleinert.  Die dicke des n-dimensionalen Würfel-graphen , 1967 .

[11]  Frank Harary,et al.  Graph Theory , 2016 .

[12]  L. W. Beineke.  Minimal decompositions of complete graphs into subgraphs with embeddability properties , 1969 .

[13]  G. Chartrand,et al.  Graphs with Forbidden Subgraphs , 1971 .

[14]  J Mayer,et al.  Décomposition de K16 en Trois Graphes Planaires , 1972 .

[15]  Paul C. Kainen,et al.  Thickness and coarseness of graphs , 1973 .

[16]  Paul C. Kainen,et al.  Some recent results in topological graph theory , 1974 .

[17]  R. K. Guy.  Combinatorics: Outerthickness and outercoarseness of graphs , 1974 .

[18]  N. Bose,et al.  Thickness of graphs with degree constrained vertices , 1977 .

[19]  Jin Akiyama,et al.  The decompositions of line graphs, middle graphs and total graphs of complete graphs into forests , 1979, Discret. Math..

[20]  Frank Harary,et al.  Covering and packing in graphs IV: Linear arboricity , 1981, Networks.

[21]  Jozef Sirán,et al.  On a Modified Concept of Thickness of a Graph , 1982 .

[22]  Anthony Mansfield,et al.  Determining the thickness of graphs is NP-hard , 1983, Mathematical Proceedings of the Cambridge Philosophical Society.

[23]  Kouhei Asano.  On the genus and thickness of graphs , 1987, J. Comb. Theory, Ser. B.

[24]  Jozef Sirán,et al.  A construction of thickness-minimal graphs , 1987, Discret. Math..

[25]  Charles J. Colbourn,et al.  Partitioning the Edges of a Planar Graph into Two Partial K-Trees , 1988 .

[26]  John H. Halton,et al.  On the thickness of graphs of given degree , 1991, Inf. Sci..

[27]  Lenwood S. Heath.  Edge coloring planar graphs with two outerplanar subgraphs , 1991, SODA '91.

[28]  Edward R. Scheinerman,et al.  On the thickness and arboricity of a graph , 1991, J. Comb. Theory, Ser. B.

[29]  Colin Cooper.  On the Trickness of Sparse Random Graphs , 1992, Comb. Probab. Comput..

[30]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[31]  Bruce A. Reed,et al.  Star arboricity , 1992, Comb..

[32]  Subramanian Ramanathan,et al.  Scheduling algorithms for multihop radio networks , 1993, TNET.

[33]  Gerhard Ringel.  Two trees in maximal planar bipartite graphs , 1993, J. Graph Theory.

[34]  Zhongfu Zhang,et al.  A short proof of Nash-Williams' theorem for the arboricity of a graph , 1994, Graphs Comb..

[35]  Robert J. Cimikowski,et al.  On Heuristics for Determining the Thickness of a Graph , 1995, Inf. Sci..

[36]  Thomas C. Shermer,et al.  On representations of some thickness-two graphs , 1995, Comput. Geom..

[37]  Vojislav Petrovic.  Decomposition of some planar graphs into trees , 1996, Discret. Math..

[38]  Chi-Kwong Li,et al.  A research problem , 1996 .

[39]  Kiran S. Kedlaya.  Outerplanar Partitions of Planar Graphs , 1996, J. Comb. Theory, Ser. B.

[40]  L. Beineke,et al.  Biplanar Graphs:: A Survey , 1997 .

[41]  Petra Mutzel,et al.  The Thickness of Graphs: A Survey , 1998, Graphs Comb..

[42]  Erkki Mäkinen,et al.  Remarks on the Thickness of a Graph , 1998, Inf. Sci..

[43]  Michael Jünger,et al.  The thickness of a minor-excluded class of graphs , 1998, Discret. Math..

[44]  Annegret Liebers,et al.  Journal of Graph Algorithms and Applications Planarizing Graphs — a Survey and Annotated Bibliography , 2022 .

[45]  Erkki Mäkinen,et al.  A genetic algorithm for determining the thickness of a graph , 2001, Inf. Sci..

[46]  A. Kaveh,et al.  An Efficient Algorithm for Embedding Nonplanar Graphs in Planes , 2002, J. Math. Model. Algorithms.

[47]  Marc Noy,et al.  Packing trees into planar graphs , 2002 .

[48]  David Eppstein,et al.  Separating Thickness from Geometric Thickness , 2002, GD.

[49]  Ruth Haas,et al.  Characterizations of arboricity of graphs , 2002, Ars Comb..

[50]  Marc Noy,et al.  Packing trees into planar graphs , 2002, J. Graph Theory.

[51]  Koichi Yamazaki,et al.  Worst case analysis of a greedy algorithm for graph thickness , 2003, Inf. Process. Lett..

[52]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[53]  David Eppstein,et al.  Testing bipartiteness of geometric intersection graphs , 2003, SODA '04.

[54]  T. Poranen.  Approximation Algorithms for Some Topological Invariants of Graphs , 2004 .

[55]  Prosenjit Bose,et al.  Partitions of Complete Geometric Graphs into Plane Trees , 2004, GD.

[56]  László A. Székely,et al.  A note on Halton's conjecture , 2004, Inf. Sci..

[57]  Gerard J. Chang,et al.  Vertex and Tree Arboricities of Graphs , 2004, J. Comb. Optim..

[58]  Daniel Gonçalves,et al.  Edge partition of planar sraphs into two outerplanar graphs , 2005, STOC '05.

[59]  Alok Aggarwal,et al.  Multilayer grid embeddings for VLSI , 2005, Algorithmica.

[60]  Ellen Gethner,et al.  Bar k-Visibility Graphs: Bounds on the Number of Edges, Chromatic Number, and Thickness , 2005, GD.

[61]  Pascal Ochem,et al.  On some arboricities in planar graphs , 2005, Electron. Notes Discret. Math..

[62]  E. Mäkinen,et al.  Remarks on the thickness and outerthickness of a graph , 2005 .

[63]  Timo Poranen,et al.  A simulated annealing algorithm for determining the thickness of a graph , 2005, Inf. Sci..

[64]  Maya Jakobine Stein,et al.  Arboricity and tree-packing in locally finite graphs , 2006, J. Comb. Theory, Ser. B.

[65]  Therese C. Biedl,et al.  Partitions of Graphs into Trees , 2006, GD.

[66]  Stefan Felsner,et al.  Thickness of Bar 1-Visibility Graphs , 2006, Graph Drawing.

[67]  David R. Wood,et al.  Graph Treewidth and Geometric Thickness Parameters , 2005, GD.

[68]  D. Gonçalves,et al.  Caterpillar arboricity of planar graphs , 2007, Discret. Math..

[69]  Raghunath Tewari,et al.  Directed Planar Reachability is in Unambiguous Log-Space , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[70]  Hal A. Kierstead,et al.  The game of arboricity , 2008, Discret. Math..

[71]  László A. Székely,et al.  Biplanar crossing numbers. II. Comparing crossing numbers and biplanar crossing numbers using the probabilistic method , 2008, Random Struct. Algorithms.

[72]  Timo Poranen,et al.  Two New Approximation Algorithms for the Maximum Planar Subgraph Problem , 2008, Acta Cybern..

[73]  Qi Liu,et al.  Tree-Thickness and Caterpillar-Thickness under Girth Constraints , 2008, Electron. J. Comb..

[74]  Ellen Gethner,et al.  Thickness‐two graphs part one: New nine‐critical graphs, permuted layer graphs, and Catlin's graphs , 2008, J. Graph Theory.

[75]  Jian-Liang Wu,et al.  The linear arboricity of planar graphs of maximum degree seven is four , 2008 .

[76]  Daqing Yang,et al.  Asymmetric marking games on line graphs , 2008, Discret. Math..

[77]  Hung-Lin Fu,et al.  The linear 3-arboricity of Kn, n and Kn , 2008, Discret. Math..

[78]  D. Gonçalves,et al.  Covering planar graphs with forests, one having a bounded maximum degree , 2008, Electron. Notes Discret. Math..

[79]  Ileana Streinu,et al.  Sparsity-certifying Graph Decompositions , 2007, Graphs Comb..

[80]  Ellen Gethner,et al.  Thickness-Two Graphs Part Two: More New Nine-Critical Graphs, Independence Ratio, Cloned Planar Graphs, and Singly and Doubly Outerplanar Graphs , 2009, Graphs Comb..

[81]  Christian A. Duncan,et al.  On Graph Thickness, Geometric Thickness, and Separator Theorems , 2011, CCCG.

[82]  Ellen Gethner,et al.  The thickness and chromatic number of r-inflated graphs , 2010, Discret. Math..

[83]  Ellen Gethner,et al.  More results on r-inflated graphs: Arboricity, thickness, chromatic number and fractional chromatic number , 2010, Ars Math. Contemp..

[84]  Jianfeng Hou,et al.  A Planar linear arboricity conjecture , 2009, J. Graph Theory.

[85]  Stéphan Thomassé,et al.  On spanning galaxies in digraphs , 2012, Discret. Appl. Math..

[86]  Yan Yang,et al.  The thickness of amalgamations of graphs , 2012, 1201.6483.

[87]  André Raspaud,et al.  Decomposing a graph into forests , 2012, J. Comb. Theory, Ser. B.

[88]  Yan Yang,et al.  A note on the thickness of Kl, m, n , 2014, Ars Comb..