Planar graphIn 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 only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
Dual graphIn the 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 pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e.
Outerplanar graphIn 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 graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors K4 and K2,3, or by their Colin de Verdière graph invariants. They have Hamiltonian cycles if and only if they are biconnected, in which case the outer face forms the unique Hamiltonian cycle. Every outerplanar graph is 3-colorable, and has degeneracy and treewidth at most 2.
Planar separator theoremIn graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of O(\sqrt{n}) vertices from an n-vertex graph (where the O invokes big O notation) can partition the graph into disjoint subgraphs each of which has at most 2n/3 vertices.
Planarity testingIn 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 be drawn in the plane without edge intersections). This is a well-studied problem in computer science for which many practical algorithms have emerged, many taking advantage of novel data structures. Most of these methods operate in O(n) time (linear time), where n is the number of edges (or vertices) in the graph, which is asymptotically optimal.
Structural engineerStructural engineers analyze, design, plan, and research structural components and structural systems to achieve design goals and ensure the safety and comfort of users or occupants. Their work takes account mainly of safety, technical, economic, and environmental concerns, but they may also consider aesthetic and social factors. Structural engineering is usually considered a specialty discipline within civil engineering, but it can also be studied in its own right.
Graphic matroidIn the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. A matroid that is both graphic and co-graphic is sometimes called a planar matroid (but this should not be confused with matroids of rank 3, which generalize planar point configurations); these are exactly the graphic matroids formed from planar graphs.
Apex graphIn 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 an apex of the graph. It is an apex, not the apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs K_5 or K_3,3, every vertex is an apex. The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove.
Graph minorIn graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges, vertices and by contracting edges. 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 bipartite graph K3,3. The Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs that is preserved by deletions and edge contractions.
Graph drawingGraph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics. A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph.
