As everyone knows :P, a graph is a CWcomplex of dimension 1. Knowing that, are there any interesting results in graph theory that arise from working with CWcomplexes? And more specifically, in algorithms / computability?
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$\begingroup$ What kind of results are you thinking about? The question is rather broad. Also, consider adding a major tag (those starting with two letters and a dot). $\endgroup$ – Sebastian Goette Nov 9 '15 at 19:07

$\begingroup$ I imagine something as "Hey, I'd like to know whether this graph has this X propery....wait! Seeing it as a CWcomplex, it's much easier!" $\endgroup$ – Guillermo Mosse Nov 9 '15 at 19:21

$\begingroup$ So I guess things like the homcomplexes used to study colorability are not what you are interested in? You only want to consider the graphs themselves as topological spaces? $\endgroup$ – Patrik Nov 9 '15 at 19:27

$\begingroup$ That sounds like something I'd be interested to read about, thanks. Are you talking about this? (link) ? Do you have any paper for me? Anyway, although what you say seems cool, yeah, my main interest is to see the graph as a complex and see what I can obtain from that $\endgroup$ – Guillermo Mosse Nov 9 '15 at 19:33

$\begingroup$ Configuration spaces in graphs are related to motionplanning problems and they are natural CWcomplexes associated to graphs. $\endgroup$ – Ryan Budney Nov 9 '15 at 19:50