279 research outputs found
Order Invariance on Decomposable Structures
Order-invariant formulas access an ordering on a structure's universe, but
the model relation is independent of the used ordering. Order invariance is
frequently used for logic-based approaches in computer science. Order-invariant
formulas capture unordered problems of complexity classes and they model the
independence of the answer to a database query from low-level aspects of
databases. We study the expressive power of order-invariant monadic
second-order (MSO) and first-order (FO) logic on restricted classes of
structures that admit certain forms of tree decompositions (not necessarily of
bounded width).
While order-invariant MSO is more expressive than MSO and, even, CMSO (MSO
with modulo-counting predicates), we show that order-invariant MSO and CMSO are
equally expressive on graphs of bounded tree width and on planar graphs. This
extends an earlier result for trees due to Courcelle. Moreover, we show that
all properties definable in order-invariant FO are also definable in MSO on
these classes. These results are applications of a theorem that shows how to
lift up definability results for order-invariant logics from the bags of a
graph's tree decomposition to the graph itself.Comment: Accepted for LICS 201
A limit law of almost -partite graphs
For integers , we study (undirected) graphs with
vertices such that the vertices can be partitioned into parts
such that every vertex has at most neighbours in its own part. The set of
all such graphs is denoted \mbP_n(l,d). We prove a labelled first-order limit
law, i.e., for every first-order sentence , the proportion of graphs
in \mbP_n(l,d) that satisfy converges as . By
combining this result with a result of Hundack, Pr\"omel and Steger \cite{HPS}
we also prove that if are integers, then
\mb{Forb}(\mcK_{1, s_1, ..., s_l}) has a labelled first-order limit law,
where \mb{Forb}(\mcK_{1, s_1, ..., s_l}) denotes the set of all graphs with
vertices , for some , in which there is no subgraph isomorphic to
the complete -partite graph with parts of sizes . In
the course of doing this we also prove that there exists a first-order formula
(depending only on and ) such that the proportion of \mcG \in
\mbP_n(l,d) with the following property approaches 1 as : there
is a unique partition of into parts such that every vertex
has at most neighbours in its own part, and this partition, viewed as an
equivalence relation, is defined by
The genus of curve, pants and flip graphs
This article is about the graph genus of certain well studied graphs in
surface theory: the curve, pants and flip graphs. We study both the genus of
these graphs and the genus of their quotients by the mapping class group. The
full graphs, except for in some low complexity cases, all have infinite genus.
The curve graph once quotiented by the mapping class group has the genus of a
complete graph so its genus is well known by a theorem of Ringel and Youngs.
For the other two graphs we are able to identify the precise growth rate of the
graph genus in terms of the genus of the underlying surface. The lower bounds
are shown using probabilistic methods.Comment: 26 pages, 9 figure
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