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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.
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 ...
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.
A planar graph cannot contain K 3,3 as a minor; an outerplanar graph cannot contain K 3,2 as a minor (These are not sufficient conditions for planarity and outerplanarity, but necessary). Conversely, every nonplanar graph contains either K 3,3 or the complete graph K 5 as a minor; this is Wagner's theorem. [9] Every complete bipartite graph.
A graph is k-outerplanar if it has a k-outerplanar embedding. [16] An outer-1-planar graph, analogously to 1-planar graphs can be drawn in a disk, with the vertices on the boundary of the disk, and with at most one crossing per edge. Every maximal outerplanar graph is a chordal graph.
A k-degenerate graph has chromatic number at most k + 1; this is proved by a simple induction on the number of vertices which is exactly like the proof of the six-color theorem for planar graphs. Since chromatic number is an upper bound on the order of the maximum clique , the latter invariant is also at most degeneracy plus one.
(That is, any graph of treewidth > k includes one of the graphs in the set as a minor.) 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]