In mathematics, **topological graph theory** is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces.^{ [1] } It also studies immersions of graphs.

Embedding a graph in a surface means that we want to draw the graph on a surface, a sphere for example, without two edges intersecting. A basic embedding problem often presented as a mathematical puzzle is the three-cottage problem. Other applications can be found in printing electronic circuits where the aim is to print (embed) a circuit (the graph) on a circuit board (the surface) without two connections crossing each other and resulting in a short circuit.

To an undirected graph we may associate an abstract simplicial complex *C* with a single-element set per vertex and a two-element set per edge.^{ [2] } The geometric realization |*C*| of the complex consists of a copy of the unit interval [0,1] per edge, with the endpoints of these intervals glued together at vertices. In this view, embeddings of graphs into a surface or as subdivisions of other graphs are both instances of topological embedding, homeomorphism of graphs is just the specialization of topological homeomorphism, the notion of a connected graph coincides with topological connectedness, and a connected graph is a tree if and only if its fundamental group is trivial.

Other simplicial complexes associated with graphs include the Whitney complex or *clique complex*, with a set per clique of the graph, and the *matching complex*, with a set per matching of the graph (equivalently, the clique complex of the complement of the line graph). The matching complex of a complete bipartite graph is called a *chessboard complex*, as it can be also described as the complex of sets of nonattacking rooks on a chessboard.^{ [3] }

John Hopcroft and Robert Tarjan ^{ [4] } derived a means of testing the planarity of a graph in time linear to the number of edges. Their algorithm does this by constructing a graph embedding which they term a "palm tree". Efficient planarity testing is fundamental to graph drawing.

Fan Chung et al.^{ [5] } studied the problem of embedding a graph into a book with the graph's vertices in a line along the spine of the book. Its edges are drawn on separate pages in such a way that edges residing on the same page do not cross. This problem abstracts layout problems arising in the routing of multilayer printed circuit boards.

Graph embeddings are also used to prove structural results about graphs, via graph minor theory and the graph structure theorem.

- ↑ J.L. Gross and T.W. Tucker, Topological graph theory, Wiley Interscience, 1987
- ↑ Graph Topology, from PlanetMath.
- ↑ Shareshian, John; Wachs, Michelle L. (2004). "Torsion in the matching complex and chessboard complex". arXiv: math.CO/0409054 .CS1 maint: multiple names: authors list (link)
- ↑ Hopcroft, John; Tarjan, Robert E. (1974). "Efficient Planarity Testing" (PDF).
*Journal of the ACM*.**21**(4): 549–568. doi:10.1145/321850.321852. hdl:1813/6011. - ↑ Chung, F. R. K.; Leighton, F. T.; Rosenberg, A. L. (1987). "Embedding Graphs in Books: A Layout Problem with Applications to VLSI Design" (PDF).
*SIAM Journal on Algebraic and Discrete Methods*.**8**(1): 33–58. doi:10.1137/0608002.

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 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.

In graph theory, a **component** of an undirected graph is an induced subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the rest of the graph. For example, the graph shown in the illustration has three components. A vertex with no incident edges is itself a component. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are also sometimes called **connected components**.

This is a **glossary of graph theory terms**. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.

In the mathematical area of graph theory, a **clique** is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.

In the mathematical area of graph theory, a **chordal graph** is one in which all cycles of four or more vertices have a *chord*, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called **rigid circuit graphs** or **triangulated graphs**.

In mathematics, topology generalizes the notion of triangulation in a natural way as follows:

In graph theory, a branch of mathematics, the **circuit rank**, **cyclomatic number**, **cycle rank**, or **nullity** of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycles, making it into a tree or forest. It is equal to the number of independent cycles in the graph. Unlike the corresponding feedback arc set problem for directed graphs, the circuit rank r is easily computed using the formula

The **graph isomorphism problem** is the computational problem of determining whether two finite graphs are isomorphic.

In the mathematical discipline of graph theory, the **dual graph** of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G 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. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.

**Geometric graph theory** in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are allowed to be arbitrary continuous curves connecting the vertices, thus it is "the theory of geometric and topological graphs". Geometric graphs are called in recent years very often spatial networks.

In graph theory, a **path decomposition** of a graph *G* is, informally, a representation of *G* as a "thickened" path graph, and the **pathwidth** of *G* is a number that measures how much the path was thickened to form *G*. More formally, a path-decomposition is a sequence of subsets of vertices of *G* such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets, and the pathwidth is one less than the size of the largest set in such a decomposition. Pathwidth is also known as **interval thickness**, **vertex separation number**, or **node searching number**.

In topological graph theory, an **embedding** of a graph on a surface is a representation of on in which points of are associated with vertices and simple arcs are associated with edges in such a way that:

In graph theory, a branch of mathematics, the **triconnected components** of a biconnected graph are a system of smaller graphs that describe all of the 2-vertex cuts in the graph. An **SPQR tree** is a tree data structure used in computer science, and more specifically graph algorithms, to represent the triconnected components of a graph. The SPQR tree of a graph may be constructed in linear time and has several applications in dynamic graph algorithms and graph drawing.

In graph theory, the **planarity testing** problem is the algorithmic problem of testing whether a given graph is a planar graph. 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, where *n* is the number of edges in the graph, which is asymptotically optimal. Rather than just being a single Boolean value, the output of a planarity testing algorithm may be a planar graph embedding, if the graph is planar, or an obstacle to planarity such as a Kuratowski subgraph if it is not.

**Clique complexes**, **flag complexes**, and **conformal hypergraphs** are closely related mathematical objects in graph theory and geometric topology that each describe the cliques of an undirected graph.

In mathematics, the **graph structure theorem** is a major result in the area of graph theory. The result establishes a deep and fundamental connection between the theory of graph minors and topological embeddings. The theorem is stated in the seventeenth of a series of 23 papers by Neil Robertson and Paul Seymour. Its proof is very long and involved. Kawarabayashi & Mohar (2007) and Lovász (2006) are surveys accessible to nonspecialists, describing the theorem and its consequences.

In combinatorial mathematics, an **Apollonian network** is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.

In graph theory, a **well-covered graph** is an undirected graph in which every minimal vertex cover has the same size as every other minimal vertex cover. Equivalently, these are the graphs in which every maximal independent set has the same size. Well-covered graphs were defined and first studied by Plummer (1970).

In graph theory, a **bipolar orientation** or ** st-orientation** of an undirected graph is an assignment of a direction to each edge that causes the graph to become a directed acyclic graph with a single source

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