In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect...

In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing...

In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar...

graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K5...

The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the complete...

In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane who...

mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each...

vertex 1 is a universal vertex. Wheel graphs are planar graphs, and have a unique planar embedding. More specifically, every wheel graph is a Halin graph. They...

In graph theory, an st-planar graph is a bipolar orientation of a plane graph for which both the source and the sink of the orientation are on the outer...

In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can...

mathematical field of graph theory, the butterfly graph (also called the bowtie graph and the hourglass graph) is a planar, undirected graph with 5 vertices...

In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called...

In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split...

terms of graph theory, by considering it in terms of constructing a graph coloring of the planar graph of adjacencies between regions. In graph-theoretic...

mathematical field of graph theory, the friendship graph (or Dutch windmill graph or n-fan) Fn is a planar, undirected graph with 2n + 1 vertices and 3n edges...

the planar separator theorem can be used to show that n-vertex planar graphs have universal graphs with O(n3/2) edges, and that bounded-degree planar graphs...

no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face (or region) so that no two faces that share...

polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected, planar graphs. The Schlegel diagram of a convex...

graph theory, a planar straight-line graph (or straight-line plane graph, or plane straight-line graph), in short PSLG, is an embedding of a planar graph...

internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs, but are more general. Any number of regions can meet at a...

theorem that planar triangle-free graphs are 3-colorable. The Grötzsch graph is a member of an infinite sequence of triangle-free graphs, each the Mycielskian...

apex 1.  An apex graph is a graph in which one vertex can be removed, leaving a planar subgraph. The removed vertex is called the apex. A k-apex graph is...

In graph theory, a Halin graph is a type of planar graph, constructed by connecting the leaves of a tree into a cycle. The tree must have at least four...

be no Hamiltonian cycle. The resulting graph is 3-connected and planar, so by Steinitz' theorem it is the graph of a polyhedron. It has 25 faces. It can...

In graph theory, the thickness of a graph G is the minimum number of planar graphs into which the edges of G can be partitioned. That is, if there exists...

In graph theory, a k-outerplanar graph is a planar graph that has a planar embedding in which the vertices belong to at most k {\displaystyle k} concentric...

In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states...

path graph is a grid graph on the 1 × n {\displaystyle 1\times n} grid. A 2 × 2 {\displaystyle 2\times 2} grid graph is a 4-cycle. Every planar graph H is...

three of the six red vertices. The Herschel graph is a polyhedral graph; this means that it is a planar graph, one that can be drawn in the plane with none...

Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. It states that a graph G is planar if and...