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%Write your title here: 
\noindent{\bf\Large Automorphism Groups of Geometrically Represented Graphs}

%List the authors as follows (you may underline the speaker, or just
%name the speaker, and mention your co-authors at the end, up to you):  
\noindent Pavel Klav\'{i}k -- {\small Charles University in Prague}

\noindent \underline{Peter Zeman} -- {\small Charles University in Prague}

\vspace{2ex}

\noindent{\bf Abstract:} %Here write your abstract

We describe a technique to determine the automorphism group of a geometrically
represented graph, by understanding the induced action of the automorphism group
on the set of all geometric representations. Each automorphism of a graph can be
decomposed into two parts: an automorphism of a representation and a morphism of
a representation to another one.  We apply this technique to interval graphs,
unit interval graphs, permutation graphs, circle graphs and comparability
graphs. We show that interval graphs have the same automorphism groups as trees
and unit interval graphs the same as disjoint unions of caterpillars. For
permutation (which are comparability graphs of the dimension two) and circle
graphs, we show their classes of automorphism groups as slightly larger than for
trees, and we give their inductive descriptions. On the other hand, we show that
any finite group is the automorphism group of a comparability graph with the
dimension at most four.

Our approach combines techniques from group theory (group products,
homomorphisms, quotients, actions) with computer science data structures
(PQ-trees, modular trees, split trees).

%^% you need a ref section, do it more or less like that : 
%^\small
%^\begin{thebibliography}{1}
%^
%^\bibitem{CCHPP}
%^R.~H. Chitnis, M.~Cygan, M.~Hajiaghayi, M.~Pilipczuk, and M.~Pilipczuk.
%^\newblock Designing {FPT} algorithms for cut problems using randomized
%^  contractions.
%^\newblock In {\em 53rd Annual {IEEE} Symposium on Foundations of Computer
%^  Science (FOCS)}, pages 460--469, 2012.
%^
%^\bibitem{Wol15}
%^P.~Wollan.
%^\newblock The structure of graphs not admitting a fixed immersion.
%^\newblock {\em Journal of Combinatorial Theory, Series {B}}, 110:47--66, 2015.

%\end{thebibliography}


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