The graph isomorphism problem on geometric graphs
Ryuhei Uehara
Abstract
The graph isomorphism (GI) problem asks whether two given graphs are
isomorphic or not. The GI problem is quite basic and simple, however,
it's time complexity is a long standing open problem. The GI problem
is clearly in NP, no polynomial time algorithm is known, and the GI
problem is not NP-complete unless the polynomial hierarchy collapses.
In this paper, we survey the computational complexity of the problem
on some graph classes that have geometric characterizations.
Sometimes the GI problem becomes polynomial time solvable when we add
some restrictions on some graph classes. The properties of these
graph classes on the boundary indicate us the essence of difficulty of
the GI problem. We also show that the GI problem is as hard as the
problem on general graphs even for grid unit intersection graphs on a
torus, that partially solves an open problem.
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