Characterizing 2-crossing-critical graphs

It is very well-known that there are precisely two minimal non-planar graphs: K 5 and K 3 , 3 (degree 2 vertices being irrelevant in this context). In the language of crossing numbers, these are the only 1-crossing-critical graphs: they each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple, 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the Mobius Ladder V 10 ; (ii) show how to obtain all the not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii) show that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of V 10 ; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of V 8 .

[1]  Louis V. Quintas,et al.  On crossing numbers and linguistic structures , 1983 .

[2]  Richard P. Vitray.  The 2 and 3 representative projective planar embeddings , 1992, J. Comb. Theory, Ser. B.

[3]  Gelasio Salazar.  Infinite families of crossing-critical graphs with given average degree , 2003, Discret. Math..

[4]  János Pach,et al.  2 Two Important Bounds and Their Applications , 1994 .

[5]  Laurent Beaudou,et al.  On the Sharpness of Some Results Relating Cuts and Crossing Numbers , 2010, Electron. J. Comb..

[6]  J. Širáň.  Additivity of the crossing number of graphs with connectivity 2 , 1984 .

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

[8]  Neil Robertson,et al.  Closed 2-cell embeddings of graphs with no V8-minors , 2001, Discret. Math..

[9]  Jesús Leaños,et al.  ON THE ADDITIVITY OF CROSSING NUMBERS OF GRAPHS , 2008 .

[10]  R. Bruce Richter,et al.  Cubic graphs with crossing number two , 1988, J. Graph Theory.

[11]  R. Bruce Richter,et al.  Crossing numbers of sequences of graphs II: Planar tiles , 2003, J. Graph Theory.

[12]  Neil Robertson,et al.  The structure of graphs not topologically containing the Wagner graph , 2016, J. Comb. Theory, Ser. B.

[13]  Robin Thomas,et al.  Large non-planar graphs and an application to crossing-critical graphs , 2011, J. Comb. Theory, Ser. B.

[14]  G. A. Dirac,et al.  CONNECTIVITY IN GRAPHS (Mathematical Expositions No.15) , 1968 .

[15]  D. W. Hall.  A note on primitive skew curves , 1943 .

[16]  J. Sheehan.  GRAPH THEORY (Encyclopedia of Mathematics and Its Applications, 21) , 1986 .

[17]  David W. Barnette.  Generating projective plane polyhedral maps , 1991, J. Comb. Theory, Ser. B.

[18]  J. Ira.  Infinite families of crossing-critical graphs with a given crossing number , 1984 .

[19]  Brendan D. McKay,et al.  Isomorph-Free Exhaustive Generation , 1998, J. Algorithms.

[20]  Carsten Thomassen,et al.  Graphs on Surfaces , 2001, Johns Hopkins series in the mathematical sciences.

[21]  Henry H. Glover,et al.  103 Graphs that are irreducible for the projective plane , 1979, J. Comb. Theory, Ser. B.

[22]  W. T. Tutte.  Graph Theory , 1984 .

[23]  Petr Hlinený,et al.  Crossing-number critical graphs have bounded path-width , 2003, J. Comb. Theory, Ser. B.

[24]  Beth Ann Austin.  2-crossing critical graphs with a V8 minor , 2012 .

[25]  Markus Chimani,et al.  On the Minimum Cut of Planarizations , 2007, Electron. Notes Discret. Math..

[26]  Martin Kochol,et al.  Construction of crossing-critical graphs , 1987, Discret. Math..

[27]  R. Bruce Richter,et al.  Crossing numbers of sequences of graphs II: Planar tiles , 2003 .

[28]  W. T. Tutte.  Connectivity in graphs , 1966 .

[29]  James A. McHugh,et al.  Algorithmic Graph Theory , 1986 .

[30]  Bojan Mohar,et al.  Obstructions for the Disk and the Cylinder Embedding Extension Problems , 1994, Combinatorics, Probability and Computing.

[31]  Dan Archdeacon,et al.  A Kuratowski theorem for the projective plane , 1981, J. Graph Theory.

[32]  R. Bruce Richter,et al.  Crossing numbers of sequences of graphs I: general tiles , 2004, Australas. J Comb..

[33]  Jozef Sirán.  Infinite families of crossing-critical graphs with a given crossing number , 1984, Discret. Math..

[34]  Drago Bokal.  Infinite families of crossing-critical graphs with prescribed average degree and crossing number , 2010, J. Graph Theory.