A graph G is planar if and only if xxx.
What can xxx be substituted for? Note that this is from a topological POV so a graph is a 1-dim cw complex and I guess the fundamental group should be used somehow.
A graph G is planar if and only if xxx.
What can xxx be substituted for? Note that this is from a topological POV so a graph is a 1-dim cw complex and I guess the fundamental group should be used somehow.
Please refer Kuratowski's theorem on Planar graphs.
Kuratowski: A graph $G$ is planar iff $G$ does not contain a sub division of $K_{5}$ or $K_{3,3}$.
There are lots of characterizations of planar graphs, e.g. Kuratowski's theorem as already mentioned. Another is Whitney's theorem that a finite graph $G$ is planar if and only if the dual matroid to the matroid of $G$ is graphic (also comes from a graph).
A very good treatment of when a graph can be embedded in the plane and more generally into other surfaces is given by the excellent book:
Graphs on Surfaces by Bojan Mohar and Carsten Thomassen (John Hopkins U. Press, 2001)