\begin{frame}{Table of contents}
  \setbeamertemplate{section in toc}[sections numbered]
  \setbeamertemplate{subsection in toc}[square]
  \tableofcontents[sections={3}]
\end{frame}

\subsection{Minimum Separating Vertex Set Heuristic}
\begin{frame}{Minimum Separating Vertex Set Heuristic}

\metroset{block=transparent}
\begin{block}{}

  \begin{columns}

    \begin{column}{0.5\textwidth}
      \begin{tikzpicture}
        \node[shape=circle,draw=black] (A) at (0,0) {$X_{j_3}$};
        \node[shape=circle,draw=black] (B) at (3,0) {$X_{j_4}$};
        \node[shape=circle,draw=black] (C) at (1.5,1.5) {$X_i$};
        \only<2->{\node[shape=circle,fill=mLightBrown] (C) at (1.5,1.5) {$X_i$};}
        \node[shape=circle,draw=black] (D) at (0,3) {$X_{j_1}$};
        \node[shape=circle,draw=black] (E) at (3,3) {$X_{j_2}$};

        \path (A) edge (C);
        \path (B) edge (C);
        \path (D) edge (C);
        \path (E) edge (C);
        \draw[style=dashed] (E) -- (3,4);
        \draw[style=dashed] (E) -- (3.8,3.8);
        \draw[style=dashed] (E) -- (4,3);
      \end{tikzpicture}
    \end{column}
    \only<3>{
      \begin{column}{0.5\textwidth}
        \begin{tikzpicture}
          \node[shape=circle,fill=mLightBrown] (C) at (0,0) {$S$};
          \node[shape=circle,fill=mLightBrown] (C1) at (-1,1) {$S \cup W_1$};
          \node[shape=circle,draw=black] (C11) at (-2,2) {$X_{j_1}$};

          \node[shape=circle,fill=mLightBrown] (C2) at (1,1) {$S \cup W_2$};
          \node[shape=circle,draw=black] (C22) at (2,2) {$X_{j_2}$};

          \node[shape=circle,fill=mLightBrown] (C3) at (-1,-1) {$S \cup W_3$};
          \node[shape=circle,draw=black] (C33) at (-2,-2) {$X_{j_3}$};

          \node[shape=circle,fill=mLightBrown] (C4) at (1,-1) {$S \cup W_4$};
          \node[shape=circle,draw=black] (C44) at (2,-2) {$X_{j_4}$};

          \draw (C) -- (C1) -- (C11);
          \draw (C) -- (C2) -- (C22);
          \draw (C) -- (C3) -- (C33);
          \draw (C) -- (C4) -- (C44);
          \draw[style=dashed] (C22) -- (2,3);
          \draw[style=dashed] (C22) -- (2.8,2.8);
          \draw[style=dashed] (C22) -- (3,2);
        \end{tikzpicture}    
      \end{column}
    }
  \end{columns}
\end{block}

\only<2>{
  \begin{block}{}
    Choose $i\in I$ such that $\vert X_i \vert$ maximal and $G[X_i]$ does not include a clique.
  \end{block}
}

\only<3>{
  \begin{block}{}
    Construct Graph $H_i$:\\
    $H_i(X_i,E_{H_i}), E_{H_i} = \{\{v,w\} \in X_i\times X_i \vert \{v,w\} \in E \vee \exists j \neq i: v,w \in X_j\}$\\
    Compute minimum separator $S$; $W_1,\dots{},W_r$ are components\\
    Construct new tree decomposition
  \end{block}
}

\end{frame}

\subsection{Other Algorithms}
\begin{frame}{Others}
  \begin{itemize}
  \item MinimalTriangulation (same principle as Minimum Separating Vertex Set Heuristic)
  \item Component Splitting
  \end{itemize}
\end{frame}

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