Oh wow! To confirm I understand this, is the following correct? "The radius-r highway dimension of a graph G is the(/a?) minimal set of vertices that must be visited when traveling a distance ∈ (r, 4r]." I'm actually a little unclear on r—I assumed HD is a function of r, but you said you iterate over all real radii to obtain HD, which would mean HD is independent of r?
HD is independent of r.
And it's not the minimal set of vertices that must be visited.. it is the smallest set of which at least one must be visited. Another good explanation of HD is in section 1.3 of the skeleton dimension paper (https://arxiv.org/pdf/1609.00512.pdf).
Oh, sorry, I was vague about the visiting—I meant the set must be visited, i.e. something in it must be visited. I'll take a look at that one too, thanks!
Yes this is helpful, thank you so much! :)