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%Write your title here: 
\noindent{\bf\Large Clique-width of Restricted Graph Classes}

%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 A.~Brandst{\"a}dt -- {\small Universit{\"a}t Rostock},

\noindent \underline{K.K. Dabrowski} -- {\small Durham University},

\noindent F.~Dross -- {\small Universit\'e de Montpellier},

\noindent S.~Huang -- {\small Simon Fraser University},

\noindent D.~Paulusma -- {\small Durham University}




\vspace{2ex}

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

The {\em clique-width} of a graph~$G$, is the minimum number of labels
needed to construct~$G$ using the following four operations:
\begin{itemize}
\item creating a new graph consisting of a single vertex~$v$ with label~$i$;
\item taking the disjoint union of two labelled graphs~$G_1$ and~$G_2$;
\item joining each vertex with label~$i$ to each vertex with label~$j$ ($i\neq j$);
\item renaming label~$i$ to~$j$.
\end{itemize}

Clique-width is of great theoretical interest because many natural algorithmic
problems that are NP-complete in general can be solved efficiently on graph
classes of bounded clique-width. This includes all problems expressible in
monadic second order logic with quantification over vertices, along with other
problems such as vertex colouring and Hamiltonian cycle. Clique-width is a
tricky parameter to deal with. Indeed, even for low values of~$c$, such as
$c=4$, we do not know if graphs of clique-width~$c$ can be detected in
polynomial time.

\bigskip
I will describe some of the tools available for dealing with clique-width and
summarize our recent work on classifying which classes of graphs have bounded
clique-width, in particular for:
$H$-free graphs~\cite{DP15}, 
$H$-free bipartite graphs~\cite{DP14},
$H$-free split graphs~\cite{BDHP15b}, 
$H$-free chordal graphs~\cite{BDHP15}, 
$H$-free weakly chordal graphs~\cite{BDHP15} and
$(H_1,H_2)$-free graphs~\cite{BDHP15,DDP15,DHP0,DP15}.


%If you need a ref section, do it more or less like that : 
\small
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%\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.


\bibitem{BDHP15}
A.~Brandst{\"a}dt, K.~K.~Dabrowski, S.~Huang, and D.~Paulusma,
\newblock Bounding the clique-width of {$H$-free} chordal graphs,
\newblock \textit{Proc. MFCS 2015, LNCS} 9235:139--150 (2015).
\newblock Full version: \textit{arXiv CoRR} abs/1502.06948 (2015).

\bibitem{BDHP15b}
A.~Brandst{\"a}dt, K.~K.~Dabrowski, S.~Huang, and D.~Paulusma,
\newblock Bounding the clique-width of {$H$-free} split graphs,
\newblock \textit{Proc. Eurocomb 2015, ENDM} (to appear).
\newblock Full version: \textit{arXiv CoRR} abs/1509.04273 (2015).

\bibitem{DDP15}
K.~K.~Dabrowski F.~Dross and D.~Paulusma,
\newblock New results on the clique-width of graph classes defined by two forbidden induced subgraphs,
\newblock (working paper).

\bibitem{DHP0}
K.~K.~Dabrowski, S.~Huang, and D.~Paulusma,
\newblock Bounding clique-width via perfect graphs,
\newblock \textit{Proc. LATA 2015, LNCS} 8977:676--688 (2015).
\newblock Full version: \textit{arXiv CoRR} abs/1406.6298 (2014).

\bibitem{DP14}
K.~K.~Dabrowski and D.~Paulusma,
\newblock Classifying the clique-width of ${H}$-free bipartite graphs,
\newblock \textit{Proc. COCOON 2014, LNCS} 8591:489--500 (2014).
\newblock Full version: \textit{Discrete Applied Mathematics} (in press).

\bibitem{DP15}
K.~K.~Dabrowski and D.~Paulusma,
\newblock Clique-width of graph classes defined by two forbidden induced subgraphs,
\newblock \textit{Proc. CIAC 2015, LNCS} 9079:167--181 (2015).
\newblock Full version: \textit{arXiv CoRR} abs/1405.7092.


\end{thebibliography}


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