[+] Shortest paths and recursive divisions in minor-free graphs.
We discuss recursive divisions and how to obtain
them in planar and minor-free graphs. This is one of the main tools
that is used to obtain a linear-time SSSP algorithm and, in fact, once
we have a suitable recursive division, the same SSSP algorithm works
for both planar and minor-closed classes. However, both the recursive
division algorithm and the SSSP analysis require the graph to be of
bounded degree and it turns out that, in general H-minor-free graphs,
a reduction to the bounded-degree case as in the planar case is not
possible. We will see how to use knitted H-partitions to overcome
this issue and obtain a generalized recursive division algorithm for
all H-minor-free classes.
[HKRS97] M. R. Henzinger, P. N. Klein, S. Rao, S. Subramanian: Faster shortest path algorithms for planar graphs. In: Journal of Computer and System Sciences, vol. 55(1):pp. 3-23, 1997.
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