Disclaimer: have not studied topology yet, currently using Wikipedia to scrap by

It just seems to me that there should be some way to connect paths in graph theory to paths (I am interested only in undirected simple graphs if it ever matters) in topology. Fix a path in some graph. Naively trying to parameterize the vertices that form the path would in general, from what I learned, result in a noncontinuous function, which is not a parameterization (thus not a topological path) by definition. Is there anyway to connect these two concepts?


1 Answer 1


You can embed any (finite) graph (simple, undirected) into $\Bbb R^3$ faithfully: vertices get mapped to points in $3$-space and every edge is a homeomorpic copy of $[0,1]$, where distinct edges only can meet at endpoints/vertices.

In such a representation of a graph $G$ as a subspace $\hat{G}$ of $\Bbb R^3$, there is a path between two vertices $v,v'$ of $G$ (in the graph sense) iff there is a path (topologically) between the corresponding $\hat{v},\hat{v'}$ in $\hat{G}$, and there is even a whole theory on computing the homotopy or homology groups of spaces of the form $\hat{G}$ in terms of matrices associated with $G$.

But paths in graphs are really the same idea as continuous paths in such (simiplicial) spaces. If you realise each edge via some $p:[0,1]\to \Bbb R^3$ then we can compose such paths to continuous new paths when the connect at a vertix (the usual double speed trick, we also see in homotopy theory).


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