  Graphs with high symmetry or regularity are the main source for
experimentally hard instances of the notoriously difficult graph isomorphism
problem. In this paper, we study the computational complexity of isomorphism
testing for line graphs of $t$-$(v,k,\lambda)$ designs. For this class of
highly regular graphs, we obtain a worst-case running time of $O(v^{\log v +
O(1)})$ for bounded parameters $t,k,\lambda$. In a first step, our approach
makes use of the Babai--Luks algorithm to compute canonical forms of
$t$-designs. In a second step, we show that $t$-designs can be reconstructed
from their line graphs in polynomial-time. The first is algebraic in nature,
the second purely combinatorial. For both, profound structural knowledge in
design theory is required. Our results extend earlier complexity results about
isomorphism testing of graphs generated from Steiner triple systems and block
designs.
