12 research outputs found
Characterizing 2-crossing-critical graphs
It is very well-known that there are precisely two minimal non-planar graphs:
and (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 M\"obius Ladder ; (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 ; and (iv) determine all the
3-connected 2-crossing-critical graphs that do not contain a subdivision of
.Comment: 176 pages, 28 figure
Characterizing 2-crossing-critical graphs
It is very well-known that there are precisely two minimal non-planar graphs: K5 and K3,3 (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 Möbius Ladder V10; (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 V10; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of V8