15 research outputs found
Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs
The Graph Isomorphism problem restricted to graphs of bounded treewidth or
bounded tree distance width are known to be solvable in polynomial time
[Bod90],[YBFT99]. We give restricted space algorithms for these problems
proving the following results: - Isomorphism for bounded tree distance width
graphs is in L and thus complete for the class. We also show that for this kind
of graphs a canon can be computed within logspace. - For bounded treewidth
graphs, when both input graphs are given together with a tree decomposition,
the problem of whether there is an isomorphism which respects the
decompositions (i.e. considering only isomorphisms mapping bags in one
decomposition blockwise onto bags in the other decomposition) is in L. - For
bounded treewidth graphs, when one of the input graphs is given with a tree
decomposition the isomorphism problem is in LogCFL. - As a corollary the
isomorphism problem for bounded treewidth graphs is in LogCFL. This improves
the known TC1 upper bound for the problem given by Grohe and Verbitsky
[GroVer06].Comment: STACS conference 2010, 12 page
The Isomorphism Problem for Planar 3-Connected Graphs is in Unambiguous Logspace
The isomorphism problem for planar graphs is known to be efficiently
solvable. For planar 3-connected graphs, the isomorphism problem can be solved
by efficient parallel algorithms, it is in the class . In this paper we
improve the upper bound for planar 3-connected graphs to unambiguous logspace,
in fact to . As a consequence of our method we get that the
isomorphism problem for oriented graphs is in . We also show that the
problems are hard for
The Complexity of Bisimulation and Simulation on Finite Systems
In this paper the computational complexity of the (bi)simulation problem over
restricted graph classes is studied. For trees given as pointer structures or
terms the (bi)simulation problem is complete for logarithmic space or NC,
respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and
S\'antha. Furthermore, if only one of the input graphs is required to be a
tree, the bisimulation (simulation) problem is contained in AC (LogCFL). In
contrast, it is also shown that the simulation problem is P-complete already
for graphs of bounded path-width
Canonizing Graphs of Bounded Tree Width in Logspace
Graph canonization is the problem of computing a unique representative, a
canon, from the isomorphism class of a given graph. This implies that two
graphs are isomorphic exactly if their canons are equal. We show that graphs of
bounded tree width can be canonized by logarithmic-space (logspace) algorithms.
This implies that the isomorphism problem for graphs of bounded tree width can
be decided in logspace. In the light of isomorphism for trees being hard for
the complexity class logspace, this makes the ubiquitous class of graphs of
bounded tree width one of the few classes of graphs for which the complexity of
the isomorphism problem has been exactly determined.Comment: 26 page
Combinatorial refinement on circulant graphs
The combinatorial refinement techniques have proven to be an efficient
approach to isomorphism testing for particular classes of graphs. If the number
of refinement rounds is small, this puts the corresponding isomorphism problem
in a low-complexity class. We investigate the round complexity of the
2-dimensional Weisfeiler-Leman algorithm on circulant graphs, i.e. on Cayley
graphs of the cyclic group , and prove that the number of rounds
until stabilization is bounded by , where is
the number of divisors of . As a particular consequence, isomorphism can be
tested in NC for connected circulant graphs of order with an odd
prime, and vertex degree smaller than .
We also show that the color refinement method (also known as the
1-dimensional Weisfeiler-Leman algorithm) computes a canonical labeling for
every non-trivial circulant graph with a prime number of vertices after
individualization of two appropriately chosen vertices. Thus, the canonical
labeling problem for this class of graphs has at most the same complexity as
color refinement, which results in a time bound of . Moreover, this provides a first example where a sophisticated approach to
isomorphism testing put forward by Tinhofer has a real practical meaning.Comment: 19 pages, 1 figur
The Weisfeiler-Leman Dimension of Planar Graphs is at most 3
We prove that the Weisfeiler-Leman (WL) dimension of the class of all finite
planar graphs is at most 3. In particular, every finite planar graph is
definable in first-order logic with counting using at most 4 variables. The
previously best known upper bounds for the dimension and number of variables
were 14 and 15, respectively.
First we show that, for dimension 3 and higher, the WL-algorithm correctly
tests isomorphism of graphs in a minor-closed class whenever it determines the
orbits of the automorphism group of any arc-colored 3-connected graph belonging
to this class.
Then we prove that, apart from several exceptional graphs (which have
WL-dimension at most 2), the individualization of two correctly chosen vertices
of a colored 3-connected planar graph followed by the 1-dimensional
WL-algorithm produces the discrete vertex partition. This implies that the
3-dimensional WL-algorithm determines the orbits of a colored 3-connected
planar graph.
As a byproduct of the proof, we get a classification of the 3-connected
planar graphs with fixing number 3.Comment: 34 pages, 3 figures, extended version of LICS 2017 pape
The Iteration Number of Colour Refinement
The Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a colouring of the vertices of its input graph.
A trivial upper bound on the iteration number of Colour Refinement on graphs of order n is n-1. We show that this bound is tight. More precisely, we prove via explicit constructions that there are infinitely many graphs G on which Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite families that we present, we show that for every natural number n ? 10, there are graphs on n vertices on which Colour Refinement requires at least n-2 iterations to reach stabilisation