Polynomial-time algorithm for Graph Isomorphism in case of Maximum Constrained Maximum Degree From Wolfram: 


  A polynomial time algorithm is however known for planar graphs (Hopcroft and Tarjan 1973, Hopcroft and Wong 1974) and when the maximum vertex degree is bounded by a constant (Luks 1982; Skiena 1990, p. 181).


What are the running times of such algorithms, in terms of size of the graphs? What algorithms are used in practice for graph isomorphism?
Algorithms used in practice differ a lot from theoretical algorithms. One popular software package is nauty. The link should also explain the algorithms used, which probably rely on combinatorial methods.

As for the running times for specific algorithms, they are likely stated in the papers. I suggest checking the references.