Topological graph theory
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-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.
Graphs as topological spacesEdit
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". Efficient planarity testing is fundamental to graph drawing.
Fan Chung et al. 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.
- 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.
- Hopcroft, John; Tarjan, Robert E. (1974). "Efficient Planarity Testing". Journal of the ACM. 21 (4): 549–568. doi:10.1145/321850.321852.
- Chung, F. R. K.; Leighton, F. T.; Rosenberg, A. L. (1987). "Embedding Graphs in Books: A Layout Problem with Applications to VLSI Design". SIAM Journal on Algebraic and Discrete Methods. 8 (1).