75,441 research outputs found
Group orderings, dynamics, and rigidity
Let G be a countable group. We show there is a topological relationship
between the space CO(G) of circular orders on G and the moduli space of actions
of G on the circle; as well as an analogous relationship for spaces of left
orders and actions on the line. In particular, we give a complete
characterization of isolated left and circular orders in terms of strong
rigidity of their induced actions of G on and R.
As an application of our techniques, we give an explicit construction of
infinitely many nonconjugate isolated points in the spaces CO(F_{2n}) of
circular orders on free groups disproving a conjecture from Baik--Samperton,
and infinitely many nonconjugate isolated points in the space of left orders on
the pure braid group P_3, answering a question of Navas. We also give a
detailed analysis of circular orders on free groups, characterizing isolated
orders
On grounded L-graphs and their relatives
We consider the graph class Grounded-L corresponding to graphs that admit an
intersection representation by L-shaped curves, where additionally the topmost
points of each curve are assumed to belong to a common horizontal line. We
prove that Grounded-L graphs admit an equivalent characterisation in terms of
vertex ordering with forbidden patterns.
We also compare this class to related intersection classes, such as the
grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance
graphs), or the outer-1-string graphs. We give constructions showing that these
classes are all distinct and satisfy only trivial or previously known
inclusions.Comment: 16 pages, 6 figure
Simultaneous Representation of Proper and Unit Interval Graphs
In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs - the simultaneous version of arguably one of the most well-studied graph classes - is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more "rigid" and therefore have less freedom in their representation. We show they can be recognized in time O(|V|*|E|) for any number of simultaneous graphs in the sunflower case where G=(V,E) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary
Groups with right-invariant multiorders
A Cayley object for a group G is a structure on which G acts regularly as a
group of automorphisms. The main theorem asserts that a necessary and
sufficient condition for the free abelian group G of rank m to have the generic
n-tuple of linear orders as a Cayley object is that m>n. The background to this
theorem is discussed. The proof uses Kronecker's Theorem on diophantine
approximation.Comment: 9 page
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