We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 many-one reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.

We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is many-one complete for several complexity classes within NC². In particular we show that tree isomorphism, when trees are encoded as strings, is NC¹-hard under AC0-reductions. NC¹-completeness thus follows from Buss's NC¹ upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is L-complete. Concerning colored graphs we show that the isomorphism problem for graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions. This result improves the existing upper bounds for the problem. We also show that the graph automorphism problem for colored graphs with color classes of size 2 is equivalent to deciding whether a graph has more than a single connected component and we prove that for color classes of size 3 the graph automorphism problem is contained in SL.

We show that the isomorphism of 3-connected planar graphs can be decided in deterministic log-space. This improves the previously known bound UL ∩ coUL of [13].

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Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NP-completeness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.

We prove that the graph isomorphism problem restricted to colored graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions. This result improves the existing upper bounds for the problem. We also

We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero. We give a deterministic polynomial time algorithm for this problem when the inputs are read-twice or read-thrice formulas. In the process, these algorithms also test if the input circuit is computing a multilinear polynomial. We further study three related computational problems on arithmetic circuits. Given an arithmetic circuit C, 1) ZMC: test if a given monomial in C has zero coefficient or not, 2) MonCount: compute the number of monomials in C, and 3) MLIN: test if C computes a multilinear polyno-mial or not. These problems were introduced by Fournier, Malod and Mengel [STACS 2012], and shown to characterize various levels of the counting hierarchy (CH). We address the above problems on read-restricted arithmetic circuits and branching programs. We prove several complexity characterizations for the above problems on these restricted classes of arithmetic circuits.

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Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs. Recently, [8] proved that planar isomorphism is complete for log-space. We extend this result further to the classes of graphs which exclude K3,3 or K5 as a minor, and give a log-space algorithm. Our algorithm decomposes K3,3 minor-free graphs into biconnected and those further into triconnected components, which are known to be either planar or K5 components [20]. This gives a triconnected component tree similar to that for planar graphs. An extension of the log-space algorithm of [8] can then be used to decide the isomorphism problem. For K5 minor-free graphs, we consider 3-connected components. These are either planar or isomorphic to the four-rung mobius ladder on 8 vertices or, with a further decomposition, one obtains planar 4-connected components [9]. We give an algorithm to get a unique decomposition of K5 minor-free graphs into bi-, tri- and 4-connected components, and construct trees, accordingly. Since the algorithm of [8] does not deal with four-connected component trees, it needs to be modified in a quite non-trivial way.