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An apex graph is a graph that may be made planar by the removal of one vertex, and a k-apex graph is a graph that may be made planar by the removal of at most k vertices. A 1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and a k-planar graph is a graph that may be drawn with at most k simple ...
An outerplanar graph (or 1-outerplanar graph) has all of its vertices on the unbounded (outside) face of the graph. A 2-outerplanar graph is a planar graph with the property that, when the vertices on the unbounded face are removed, the remaining vertices all lie on the newly formed unbounded face. And so on.
Each of these sets of forbidden minors includes at least one planar graph. For k = 1, the unique forbidden minor is a 3-vertex cycle graph. [5] For k = 2, the unique forbidden minor is the 4-vertex complete graph K 4. [5] For k = 3, there are four forbidden minors: K 5, the graph of the octahedron, the pentagonal prism graph, and the Wagner graph.
A graph is k-vertex-connected, but not necessarily planar, if and only if it has a convex embedding into (k −1)-dimensional space in which an arbitrary k-tuple of vertices are placed at the vertices of a simplex and, for each remaining vertex v, the convex hull of the neighbors of v is full-dimensional with v in its interior.
That is, if there exists a collection of k planar graphs, all having the same set of vertices, such that the union of these planar graphs is G, then the thickness of G is at most k. [1] [2] In other words, the thickness of a graph is the minimum number of planar subgraphs whose union equals to graph G. [3] Thus, a planar graph has thickness one.
K 1 through K 4 are all planar graphs. However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph K 5 plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K 5 nor the complete bipartite ...
Kuratowski's theorem that a graph is planar if and only if it does not contain a subgraph that is a subdivision of K 5 (the complete graph on five vertices) or K 3,3 (the utility graph, a complete bipartite graph on six vertices, three of which connect to each of the other three).
[1] [6] This parameter k is known as the skewness of the graph. [3] [7] There has also been some study of a related problem, finding the largest planar induced subgraph of a given graph. Again, this is NP-hard, but fixed-parameter tractable when all but a few vertices belong to the induced subgraph. [8]