12 research outputs found
A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded Genus
The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension can also be characterised in terms of the number of variables that is required to describe the graph up to isomorphism in first-order logic with counting quantifiers.
It is known that the WL dimension is upper-bounded for all graphs that exclude some fixed graph as a minor [M. Grohe, 2017]. However, the bounds that can be derived from this general result are astronomic. Only recently, it was proved that the WL dimension of planar graphs is at most 3 [S. Kiefer et al., 2017].
In this paper, we prove that the WL dimension of graphs embeddable in a surface of Euler genus g is at most 4g+3. For the WL dimension of graphs embeddable in an orientable surface of Euler genus g, our approach yields an upper bound of 2g + 3
Graphs Identified by Logics with Counting
We classify graphs and, more generally, finite relational structures that are
identified by C2, that is, two-variable first-order logic with counting. Using
this classification, we show that it can be decided in almost linear time
whether a structure is identified by C2. Our classification implies that for
every graph identified by this logic, all vertex-colored versions of it are
also identified. A similar statement is true for finite relational structures.
We provide constructions that solve the inversion problem for finite
structures in linear time. This problem has previously been shown to be
polynomial time solvable by Martin Otto. For graphs, we conclude that every
C2-equivalence class contains a graph whose orbits are exactly the classes of
the C2-partition of its vertex set and which has a single automorphism
witnessing this fact.
For general k, we show that such statements are not true by providing
examples of graphs of size linear in k which are identified by C3 but for which
the orbit partition is strictly finer than the Ck-partition. We also provide
identified graphs which have vertex-colored versions that are not identified by
Ck.Comment: 33 pages, 8 Figure
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