2,589 research outputs found
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
Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces
The Subgraph Isomorphism problem asks, given a host graph G on n vertices and
a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P.
The restriction of this problem to planar graphs has often been considered.
After a sequence of improvements, the current best algorithm for planar graphs
is a linear time algorithm by Dorn (STACS '10), with complexity .
We generalize this result, by giving an algorithm of the same complexity for
graphs that can be embedded in surfaces of bounded genus. At the same time, we
simplify the algorithm and analysis. The key to these improvements is the
introduction of surface split decompositions for bounded genus graphs, which
generalize sphere cut decompositions for planar graphs. We extend the algorithm
for the problem of counting and generating all subgraphs isomorphic to P, even
for the case where P is disconnected. This answers an open question by Eppstein
(SODA '95 / JGAA '99)
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