Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable

Lubiw showed that several variants of Graph Isomorphism are NPcomplete, where the solutions are required to satisfy certain additional constraints [14]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs X1 = (V,E1) and X2 = (V,E2) and a subset R ⊆ V × V of forbidden pairs whether there is an isomorphism π from X1 to X2 such that i π 6= j for all (i, j) ∈ R. We prove that this problem and several of its generalizations are in fact in FPT: • The problem of deciding whether there is an isomorphism between two graphs that moves k vertices and satisfies Lubiw-style constraints is in FPT, with k and |R| as parameters. The problem remains in FPT even if a CNF of such constraints is allowed. As a consequence of the main result it follows that the problem to decide whether there is an isomorphism that moves exactly k vertices is in FPT. This solves a question left open in [2]. • When the weight and complexity are unrestricted, finding isomorphisms that satisfy a CNF of Lubiw-style constraints is in FPT. • Checking if there is an isomorphism between two graphs that has complexity t is also in FPT with t as parameter, where the complexity of a permutation π is the Cayley measure defined as the minimum number t such that π can be expressed as a product of t transpositions. • We consider a more general problem in which the vertex set of a graph X is partitioned into Red and Blue, and we are interested in an automorphism that stabilizes Red and Blue and moves exactly k vertices in Blue, where k is the parameter. This problem was introduced in [6], and in [2] we showed that it is W[1]-hard even with color classes of size 4 inside Red. Now, for color classes of size at most 3 inside Red, we show the problem is in FPT. In the non-parameterized setting, all these problems are NP-complete. Also, they all generalize in several ways the problem to decide whether there is an isomorphism between two graphs that moves at most k vertices, shown to be in FPT by Schweitzer [15]. ∗An extended abstract of this article appears in the proceedings of IPEC 2017. 1 ar X iv :1 70 9. 10 06 3v 1 [ cs .C C ] 2 8 Se p 20 17