The crossing number of a graph @math is the least number of crossings over all possible drawings of @math . We present a structural characterization of graphs with crossing number one.
Akka, Jendrol, Kle s c , and Panshetty @cite_15 obtained a characterization of planar graphs whose line graph has crossing number two.
@cite_15
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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 . In this paper we deduce a necessary and sufficient condition for a line grah to have crossing number 1. In addition, we prove that the line graph of any nonplanar graph has crossing number greater than 2. It is proved that every cubic graph with crossing number at least two contains a subdivision of one of eight graphs. We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K"4","k, the graph obtained from a cycle of length 2k+1 by adding an edge joining every pair of vertices at distance exactly k, or the graph obtained from a cycle of length k by adding two vertices adjacent to each other and to every vertex on the cycle. We also prove a version of this for subdivisions rather than minors, and relax the connectivity to allow 3-cuts with one side planar and of bounded size. We deduce that for every integer k there are only finitely many 3-connected 2-crossing-critical graphs with no subdivision isomorphic to the graph obtained from a cycle of length 2k by joining all pairs of diagonally opposite vertices.