2,151 research outputs found
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four
Automorphism Groups of Geometrically Represented Graphs
Interval graphs are intersection graphs of closed intervals and circle graphs are intersection graphs of chords of a circle. We study automorphism groups of these graphs. We show that interval graphs have the same automorphism groups as trees, and circle graphs have the same
as pseudoforests, which are graphs with at most one cycle in every connected component.
Our technique determines automorphism groups for classes with a
strong structure of all geometric representations, and it can be applied to other graph classes. Our results imply polynomial-time algorithms for computing automorphism groups in term of group products
The automorphism group of the free group of rank two is a CAT(0) group
We prove that the automorphism group of the braid group on four strands acts
faithfully and geometrically on a CAT(0) 2-complex. This implies that the
automorphism group of the free group of rank two acts faithfully and
geometrically on a CAT(0) 2-complex, in contrast to the situation for rank
three and above.Comment: 7 pages, 2 figures. The manuscript has been modified in minor ways in
accordance with a referee's recommendations, and a misattribution of the
result "Aut F_2 is biautomatic" has been correcte
Automorphism Groups and Adversarial Vertex Deletions
Any finite group can be encoded as the automorphism group of an unlabeled
simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated
a construction that allows any ordered pair of finite groups to be represented
as the automorphism group of a graph and a vertex-deleted subgraph. In this
note, we describe a generalized scenario as a game between a player and an
adversary: An adversary provides a list of finite groups and a number of
rounds. The player constructs a graph with automorphism group isomorphic to the
first group. In the following rounds, the adversary selects a group and the
player deletes a vertex such that the automorphism group of the corresponding
vertex-deleted subgraph is isomorphic to the selected group. We provide a
construction that allows the player to appropriately respond to any sequence of
challenges from the adversary.Comment: 5 page
Hurwitz equivalence of braid monodromies and extremal elliptic surfaces
We discuss the equivalence between the categories of certain ribbon graphs
and subgroups of the modular group and use it to construct
exponentially large families of not Hurwitz equivalent simple braid monodromy
factorizations of the same element. As an application, we also obtain
exponentially large families of {\it topologically} distinct algebraic objects
such as extremal elliptic surfaces, real trigonal curves, and real elliptic
surfaces
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