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...

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...

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...

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...

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

path of two adjacent edges. In Vizing's planar graph conjecture, Vizing (1965) states that all simple, planar graphs with maximum degree six or seven are...

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...

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...

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...

wheel graphs. A maximal planar graph is a planar graph in which all faces, even the outer face, are triangles. A maximal planar graph is node-pancyclic if...

planar structures. However, this characterization, the Geiringer–Laman theorem, had already been discovered in 1927 by Hilda Geiringer. Laman graphs arise...

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 a graph that...

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...

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...

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

vertices in the whole graph. Theorem—A 4-connected planar triangulation has a Hamiltonian cycle. Theorem—A 4-connected planar graph has a Hamiltonian cycle...

a graph embedding in the plane. The impossibility of the puzzle corresponds to the fact that K 3 , 3 {\displaystyle K_{3,3}} is not a planar graph. Multiple...

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 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 book embedding is a generalization of planar embedding of a graph to embeddings in a book, a collection of half-planes all having the...

vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph. A planar graph is a graph whose vertices and edges can...

In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space ⁠ R n {\displaystyle \mathbb {R}...

G. A maximal planar graph G is a finite simple planar graph to which no more edges can be added while preserving planarity. Such a graph always has a...

every wheel graph is a Halin graph. They are self-dual: the planar dual of any wheel graph is an isomorphic graph. Every maximal planar graph, other than...

string graphs was eventually proven to be NP-complete, implying that no simple characterization is likely to exist. Every planar graph is a string graph: one...

unsatisfiable. Like 3SAT, PLANAR-SAT is NP-complete, and is commonly used in reductions. Every 3SAT problem can be converted to an incidence graph in the following...

and to find graph drawings with high angular resolution. Both the chromatic number and the degeneracy of the kth power of a planar graph of maximum degree...

king's graph, the graph of moves of a chess king on a chessboard, which can be constructed as a strong product of path graphs. Decompositions of planar graphs...

known that any finite graph can be embedded in 3-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} . A planar graph is one that can be embedded...