A classic example, the Four Colour Theorem, illustrates a fundamental property of planar graphs, while recent advancements have extended these ideas to more specialised variants, including list ...

Now imagine that you want to insert a new edge connecting two nodes in a planar graph, say nodes 1 and 6 in the example below. To do so, you’re going to perform a series of flips. From the starting ...

Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more ...

If G is a planar graph, we may add edges to construct a maximal planar graph H containing G, so that H triangulates the sphere. If G is toroidal, then by adding edges we can extend G to a maximal ...

"This allows us to exploit the exact, efficient solutions for planar spin glasses. For non-planar graphs, such solutions are computationally intractable," explained the researchers.

It is easy to check in quadratic time if a graph is 1-apex - simply test for each vertex v, if G - v is planar. Problem 1: Find a subquadratic algorithm for testing if a given graph is 1-planar. It is ...