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 a planar embedding of the graph.
Kuratowski's theoremIn graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5{\displaystyle K_{5}} (the complete graph on five vertices) nor of K3,3{\displaystyle K_{3,3}} (a complete bipartite graph on six vertices, three of which connect to each of the other three, also known as the utility graph).Graph minorIn graph theory, an undirected graph H is called a minor of the undirected graph G if H can be formed from G by deleting edges and 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.Characterization (mathematics)In mathematics, a characterization of an object is a set of conditions that, while possibly different from the definition of the object, is logically equivalent to it. To say that "Property P characterizes object X" is to say that not only does X have propertyP, but that X is the only thing that has property P (i.e., P is a defining property of X). Similarly, a set of properties P is said to characterize X, when these properties distinguish X from all other objects.
HypergraphGeneralization of graph theoryAn example of an undirected hypergraph, with X={v1,v2,v3,v4,v5,v6,v7}{\displaystyle X=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\}} and E={e1,e2,e3,e4}={\displaystyle E=\{e_{1},e_{2},e_{3},e_{4}\}=}{{v1,v2,v3},{\displaystyle \{\{v_{1},v_{2},v_{3}\},}{v2,v3},{\displaystyle \{v_{2},v_{3}\},}{v3,v5,v6},{\displaystyle \{v_{3},v_{5},v_{6}\},}{v4}}{\displaystyle \{v_{4}\}\}}. This hypergraph has order 7 and size 4.Homeomorphism (graph theory)In graph theory, two graphs G {\displaystyle G} and G ′ {\displaystyle G'} are homeomorphic if there is a graph isomorphism from some subdivision of G {\displaystyle G} to some subdivision of G ′ {\displaystyle G'} .
Complete graphIn the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg.
Complete bipartite graphIn the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg.
Graph (discrete mathematics)In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line).
Graph theoryArea of discrete mathematicsA graph with 6 vertices and 7 edgesIn mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called arcs, links, or lines).
Glossary of graph theoryLook up in Wiktionary, the free dictionary.
If and only ifIn logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") states that the truth values of two statements are equal. It is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false.*As an Amazon Associate I earn from qualifying purchases.