5 research outputs found
Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth
We give a fixed-parameter tractable algorithm that, given a parameter and
two graphs , either concludes that one of these graphs has treewidth
at least , or determines whether and are isomorphic. The running
time of the algorithm on an -vertex graph is ,
and this is the first fixed-parameter algorithm for Graph Isomorphism
parameterized by treewidth.
Our algorithm in fact solves the more general canonization problem. We namely
design a procedure working in time that, for a
given graph on vertices, either concludes that the treewidth of is
at least , or: * finds in an isomorphic-invariant way a graph
that is isomorphic to ; * finds an isomorphism-invariant
construction term --- an algebraic expression that encodes together with a
tree decomposition of of width .
Hence, the isomorphism test reduces to verifying whether the computed
isomorphic copies or the construction terms for and are equal.Comment: Full version of a paper presented at FOCS 201
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