22,136 research outputs found
Universal graphs and universal permutations
Let be a family of graphs and the set of -vertex graphs in .
A graph containing all graphs from as induced subgraphs is
called -universal for . Moreover, we say that is a proper
-universal graph for if it belongs to . In the present paper, we
construct a proper -universal graph for the class of split permutation
graphs. Our solution includes two ingredients: a proper universal 321-avoiding
permutation and a bijection between 321-avoiding permutations and symmetric
split permutation graphs. The -universal split permutation graph constructed
in this paper has vertices, which means that this construction is
order-optimal.Comment: To appear in Discrete Mathematics, Algorithms and Application
Small Superpatterns for Dominance Drawing
We exploit the connection between dominance drawings of directed acyclic
graphs and permutations, in both directions, to provide improved bounds on the
size of universal point sets for certain types of dominance drawing and on
superpatterns for certain natural classes of permutations. In particular we
show that there exist universal point sets for dominance drawings of the Hasse
diagrams of width-two partial orders of size O(n^{3/2}), universal point sets
for dominance drawings of st-outerplanar graphs of size O(n\log n), and
universal point sets for dominance drawings of directed trees of size O(n^2).
We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}),
riffle permutations (321-, 2143-, and 2413-avoiding permutations) have
superpatterns of size O(n), and the concatenations of sequences of riffles and
their inverses have superpatterns of size O(n\log n). Our analysis includes a
calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of
the 321-superpattern siz
Superpatterns and Universal Point Sets
An old open problem in graph drawing asks for the size of a universal point
set, a set of points that can be used as vertices for straight-line drawings of
all n-vertex planar graphs. We connect this problem to the theory of
permutation patterns, where another open problem concerns the size of
superpatterns, permutations that contain all patterns of a given size. We
generalize superpatterns to classes of permutations determined by forbidden
patterns, and we construct superpatterns of size n^2/4 + Theta(n) for the
213-avoiding permutations, half the size of known superpatterns for
unconstrained permutations. We use our superpatterns to construct universal
point sets of size n^2/4 - Theta(n), smaller than the previous bound by a 9/16
factor. We prove that every proper subclass of the 213-avoiding permutations
has superpatterns of size O(n log^O(1) n), which we use to prove that the
planar graphs of bounded pathwidth have near-linear universal point sets.Comment: GD 2013 special issue of JGA
Infinite quantum permutations
We define and study quantum permutations of infinite sets. This leads to
discrete quantum groups which can be viewed as infinite variants of the quantum
permutation groups introduced by Wang. More precisely, the resulting quantum
groups encode universal quantum symmetries of the underlying sets among all
discrete quantum groups. We also discuss quantum automorphisms of infinite
graphs, including some examples and open problems regarding both the existence
and non-existence of quantum symmetries in this setting.Comment: 27 page
An explicit universal cycle for the (n-1)-permutations of an n-set
We show how to construct an explicit Hamilton cycle in the directed Cayley
graph Cay({\sigma_n, sigma_{n-1}} : \mathbb{S}_n), where \sigma_k = (1 2 >...
k). The existence of such cycles was shown by Jackson (Discrete Mathematics,
149 (1996) 123-129) but the proof only shows that a certain directed graph is
Eulerian, and Knuth (Volume 4 Fascicle 2, Generating All Tuples and
Permutations (2005)) asks for an explicit construction. We show that a simple
recursion describes our Hamilton cycle and that the cycle can be generated by
an iterative algorithm that uses O(n) space. Moreover, the algorithm produces
each successive edge of the cycle in constant time; such algorithms are said to
be loopless
Infinite primitive directed graphs
A group G of permutations of a set Ω is primitive if it acts transitively on Ω, and the only G-invariant equivalence relations on Ω are the trivial and universal relations.
A digraph Γ is primitive if its automorphism group acts primitively on its vertex set, and is infinite if its vertex set is infinite. It has connectivity one if it is connected and there exists a vertex α of Γ, such that the induced digraph Γ∖{α} is not connected. If Γ has connectivity one, a lobe of Γ is a connected subgraph that is maximal subject to the condition that it does not have connectivity one. Primitive graphs (and thus digraphs) with connectivity one are necessarily infinite.
The primitive graphs with connectivity one have been fully classified by Jung and Watkins: the lobes of such graphs are primitive, pairwise-isomorphic and have at least three vertices. When one considers the general case of a primitive digraph with connectivity one, however, this result no longer holds. In this paper we investigate the structure of these digraphs, and obtain a complete characterisation
On the Expressive Power of Geometric Graph Neural Networks
The expressive power of Graph Neural Networks (GNNs) has been studied
extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However,
standard GNNs and the WL framework are inapplicable for geometric graphs
embedded in Euclidean space, such as biomolecules, materials, and other
physical systems. In this work, we propose a geometric version of the WL test
(GWL) for discriminating geometric graphs while respecting the underlying
physical symmetries: permutations, rotation, reflection, and translation. We
use GWL to characterise the expressive power of geometric GNNs that are
invariant or equivariant to physical symmetries in terms of distinguishing
geometric graphs. GWL unpacks how key design choices influence geometric GNN
expressivity: (1) Invariant layers have limited expressivity as they cannot
distinguish one-hop identical geometric graphs; (2) Equivariant layers
distinguish a larger class of graphs by propagating geometric information
beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable
maximally powerful geometric GNNs; and (4) GWL's discrimination-based
perspective is equivalent to universal approximation. Synthetic experiments
supplementing our results are available at
https://github.com/chaitjo/geometric-gnn-dojoComment: NeurIPS 2022 Workshop on Symmetry and Geometry in Neural
Representation
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
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