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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 ...
The theory of topological graphs is an area of graph theory, mainly concerned with combinatorial properties of topological graphs, in particular, with the crossing patterns of their edges. It is closely related to graph drawing , a field which is more application oriented, and topological graph theory , which focuses on embeddings of graphs in ...
In algebraic topology and graph theory, graph homology describes the homology groups of a graph, where the graph is considered as a topological space. It formalizes the idea of the number of "holes" in the graph. It is a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1 ...
This is in analogy to the Petersen family, which too is named after its member the Petersen graph. The Heawood families are significant in topological graph theory. They contain the smallest known examples of intrinsically knotted graphs, [1] of graphs that are not 4-flat, and of graphs with Colin de Verdière graph invariant =.
Download as PDF; Printable version; In other projects ... move to sidebar hide. Help. Topological graph theory is a branch of graph theory. Its main topic is the ...
The associated topological space of a graph is connected (with respect to the graph topology) if and only if the original graph is connected.; Every connected graph contains at least one maximal tree , that is, a tree that is maximal with respect to the order induced by set inclusion on the subgraphs of which are trees.
A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three-dimensional analogue of the planar graphs. [1] Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding. Flat embeddings are automatically linkless, but not vice versa. [2]
The Euler genus is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps or on a sphere with n/2 handles. [5] In topological graph theory there are several definitions of the genus of a group. Arthur T. White introduced the following concept.