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Concept
Topological graph theory
Summary
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. 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 utilities 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.
Graph (topology) and Graph homology
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. 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.
John Hopcroft and Robert Tarjan 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".
Official source
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