On the Parallel Parameterized Complexity of the Graph Isomorphism Problem
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In this paper, we study the parallel and the space complexity of the graph isomorphism problem () for several parameterizations. Let H={H_1,H_2,...,H_l} be a finite set of graphs where |V(H_i)|≤ d for all i and for some constant d. Let G be an H-free graph class i.e., none of the graphs G∈G contain any H ∈H as an induced subgraph. We show that parameterized by vertex deletion distance to G is in a parameterized version of ^1, denoted -^1, provided the colored graph isomorphism problem for graphs in G is in ^1. From this, we deduce that parameterized by the vertex deletion distance to cographs is in -^1. The parallel parameterized complexity of parameterized by the size of a feedback vertex set remains an open problem. Towards this direction we show that the graph isomorphism problem is in -^0 when parameterized by vertex cover or by twin-cover. Let G' be a graph class such that recognizing graphs from G' and the colored version of for G' is in logspace (Ł). We show that for bounded vertex deletion distance to G' is in Ł. From this, we obtain logspace algorithms for for graphs with bounded vertex deletion distance to interval graphs and graphs with bounded vertex deletion distance to cographs.
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