%%LaTeX

\chapter{Third Step}
%%\chapter*{3. Third Step}

%%\addcontentsline{toc}{chapter}{3. Third Step \dotfill}

Here is an example of a picture that spread in one page,
the code of this picture is written in pstricks.
To do this, you need to include the following lines in your header file.

\begin{verbatim}
  \usepackage{pstricks}
  \usepackage{pst-node}
\end{verbatim}

The following is Figure \ref{F:compEx}.

\begin{center}
  \begin{figure}[htbp]
  \scalebox{0.8}{
    \subfigure[Graph $G$.]{
      \begin{pspicture}(-3, -5.5)(3, 0.5)
        \psset{nodesep=0pt, shortput=nab, npos=0.5, labelsep=2pt, arrowsize=0.2}
          
        \cnode*(0, 0){0.16cm}{a}
        \cnode*(-1.2, -1.2){0.16cm}{b}
        \cnode*(1.2, -1.2){0.16cm}{c}
        \cnode*(-2.4, -2.4){0.16cm}{d}
        \cnode*(2.4, -2.4){0.16cm}{e}
        \cnode*(-1.2, -3.6){0.16cm}{f}
        \cnode*(1.2, -3.6){0.16cm}{g}
        \cnode*(2.4, -4.8){0.16cm}{h}

        \nput{90}{a}{$a$}
        \nput{90}{b}{$b$}
        \nput{90}{c}{$c$}
        \nput{180}{d}{$d$}
        \nput{0}{e}{$e$}
        \nput{-90}{f}{$f$}
        \nput{-90}{g}{$g$}
        \nput{-90}{h}{$h$}
     
        \ncline[linewidth=0.08cm, linecolor=black]{a}{b}
        \ncline[linewidth=0.08cm, linecolor=black]{a}{c}
        \ncline[linewidth=0.08cm, linecolor=black]{b}{d}
        \ncline[linewidth=0.08cm, linecolor=black]{b}{f}
        \ncline[linewidth=0.08cm, linecolor=black]{c}{d}
        \ncline[linewidth=0.08cm, linecolor=black]{c}{e}
        \ncline[linewidth=0.08cm, linecolor=black]{c}{g}
        \ncline[linewidth=0.08cm, linecolor=black]{d}{f}
        \ncline[linewidth=0.08cm, linecolor=black]{e}{h}
        \ncline[linewidth=0.08cm, linecolor=black]{g}{h}

      \end{pspicture}
    }

    \hspace*{1cm} 
    \subfigure[A tree decompostion of $G$.]{ 
      %%\scalebox{0.9}{
      \begin{pspicture}(-2, -4)(2, 0.5)
        \psset{nodesep=0pt, shortput=nab, npos=0.5, labelsep=2pt, arrowsize=0.2}
  
        \rput(0, 0){\rnode{abc}{\psovalbox[fillstyle=solid, fillcolor=white]{$a,\, b,\, c$}}} 
        \rput(-1.2, -1.2){\rnode{bcd}{\psovalbox[fillstyle=solid, fillcolor=white]{$b,\, c,\, d$}}} 
        %%\rput(1.2, -1.2){\rnode{bce}{\psovalbox[fillstyle=solid, fillcolor=white]{$b,\, c,\, e$}}} 
        \rput(-1.2, -2.4){\rnode{bdf}{\psovalbox[fillstyle=solid, fillcolor=white]{$b,\, d,\, f$}}} 
        \rput(1.2, -1.2){\rnode{ceg}{\psovalbox[fillstyle=solid, fillcolor=white]{$c,\, e,\, g$}}} 
        \rput(1.2, -2.4){\rnode{egh}{\psovalbox[fillstyle=solid, fillcolor=white]{$e,\, g,\, h$}}} 
       
        \ncline[linewidth=0.08cm, linecolor=black]{abc}{bcd}
        \ncline[linewidth=0.08cm, linecolor=black]{abc}{ceg}
        \ncline[linewidth=0.08cm, linecolor=black]{bcd}{bdf}
        \ncline[linewidth=0.08cm, linecolor=black]{ceg}{egh}
      \end{pspicture}
      %%}
    }

    \hspace*{1cm} 
    \subfigure[Another tree decompostion of $G$.]{ 
      %%\scalebox{0.9}{
      \begin{pspicture}(-2, -4)(2, 0.5)
        \psset{nodesep=0pt, shortput=nab, npos=0.5, labelsep=2pt, arrowsize=0.2}
  
        \rput(0, 0){\rnode{abc}{\psovalbox[fillstyle=solid, fillcolor=white]{$a,\, b,\, c$}}} 
        \rput(0, -1.2){\rnode{bcd}{\psovalbox[fillstyle=solid, fillcolor=white]{$b,\, c,\, d$}}} 
        \rput(-1.2, -2.4){\rnode{bdf}{\psovalbox[fillstyle=solid, fillcolor=white]{$b,\, d,\, f$}}} 
        \rput(1.2, -2.4){\rnode{ceg}{\psovalbox[fillstyle=solid, fillcolor=white]{$c,\, e,\, g$}}} 
        \rput(1.2, -3.6){\rnode{egh}{\psovalbox[fillstyle=solid, fillcolor=white]{$e,\, g,\, h$}}} 
       
        \ncline[linewidth=0.08cm, linecolor=black]{abc}{bcd}
        \ncline[linewidth=0.08cm, linecolor=black]{bcd}{bdf}
        \ncline[linewidth=0.08cm, linecolor=black]{bcd}{ceg}
        \ncline[linewidth=0.08cm, linecolor=black]{ceg}{egh}
      \end{pspicture}
    %%}
    }
  } %%\scalebox
  \caption{A graph of bounded treewidth 2 and two of its tree decompositions.}
  \label{F:compEx}
  \end{figure}
\end{center}


Figure \ref{F:solEx} is separted in the next page (because it is large).

\begin{figure}[htbp]
  \begin{center}
  \scalebox{0.8}{
    \subfigure[An optical network $N$ of bounded treewidth 2, where $a$, $b$, and $c$ are $C$-nodes.]{
    \begin{pspicture}(0, -3.5)(6.5, 3.5)
      \psset{nodesep=0pt, shortput=nab, npos=0.5, labelsep=2pt, arrowsize=0.2}
        
      \cvnode{3}{2.8}{0.72}{a}{$a$}
      \cvnode{1.5}{1.2}{0.72}{b}{$b$}
      \cvnode{4.5}{1.2}{0.72}{c}{$c$}
      \ncnode{0}{0}{0.4}{d}{$d$}
      \ncnode{6}{0}{0.4}{e}{$e$}
      \ncnode{1.5}{-1.2}{0.4}{f}{$f$}
      \ncnode{4.5}{-1.2}{0.4}{g}{$g$}
      \ncnode{6}{-2.8}{0.4}{h}{$h$}
  
      \ncline[linewidth=0.08cm, linecolor=black]{a}{b}_{$1$}
      \ncline[linewidth=0.08cm, linecolor=black]{a}{b}_{$1$}
      \ncline[linewidth=0.08cm, linecolor=black]{a}{c}^{$2$}
      \ncline[linewidth=0.08cm, linecolor=black]{b}{d}_{$1$}
      \psset{npos=0.7}
      \ncline[linewidth=0.08cm, linecolor=black]{b}{f}^{$3$}
      \psset{npos=0.3}
      \ncline[linewidth=0.08cm, linecolor=black]{c}{d}^{$4$}
      \psset{npos=0.5}
      \ncline[linewidth=0.08cm, linecolor=black]{c}{e}^{$2$}
      \ncline[linewidth=0.08cm, linecolor=black]{c}{g}_{$4$}
      \ncline[linewidth=0.08cm, linecolor=black]{d}{f}_{$3$}
      \ncline[linewidth=0.08cm, linecolor=black]{e}{h}^{$2$}
      \ncline[linewidth=0.08cm, linecolor=black]{g}{h}_{$4$}

   \end{pspicture}
    }

    \hspace*{2cm}
    \subfigure[Node $c$ dominates $a$, $d$, $e$, $g$, $h$ through the non-gray paths.]{
    \begin{pspicture}(-0.5, -3.5)(6.5, 3.5)
      \psset{nodesep=0pt, shortput=nab, npos=0.5, labelsep=2pt, arrowsize=0.2}
        
      \cvnode{3}{2.8}{0.72}{a}{$a$}
      \graycvnode{1.5}{1.2}{0.72}{b}{$b$}
      \cvnodefill{4.5}{1.2}{0.72}{c}{$c$}
      \ncnode{0}{0}{0.4}{d}{$d$}
      \ncnode{6}{0}{0.4}{e}{$e$}
      \grayncnode{1.5}{-1.2}{0.4}{f}{$f$}
      \ncnode{4.5}{-1.2}{0.4}{g}{$g$}
      \ncnode{6}{-2.8}{0.4}{h}{$h$}
  
      \ncline[linewidth=0.08cm, linecolor=gray]{a}{b}_{\gray $1$}
      \ncline[linewidth=0.16cm, linecolor=black]{a}{c}^{$2$}
      \ncline[linewidth=0.08cm, linecolor=gray]{b}{d}_{\gray $1$}
      \psset{npos=0.7}
      \ncline[linewidth=0.08cm, linecolor=gray]{b}{f}^{\gray $3$}
      \psset{npos=0.3}
      \ncline[linewidth=0.16cm, linecolor=black]{c}{d}^{$4$}
      \psset{npos=0.5}
      \ncline[linewidth=0.16cm, linecolor=black]{c}{e}^{$2$}
      \ncline[linewidth=0.16cm, linecolor=black]{c}{g}_{$4$}
      \ncline[linewidth=0.08cm, linecolor=gray]{d}{f}_{\gray $3$}
      \ncline[linewidth=0.16cm, linecolor=black]{e}{h}^{$2$}
      \ncline[linewidth=0.16cm, linecolor=black]{g}{h}_{$4$}
    \end{pspicture}
    } % subfigure
  } % scalebox 

  \scalebox{0.8}{
    \subfigure[Nodes $a$, $b$, $d$, and $f$ can send messages to each other;
               so are $a$, $c$, $d$, $e$, $g$, and $h$.
               However, $b$ cannot send messages to $c$.]{
    \begin{pspicture}(0, -3.5)(6.5, 3.5)
      \psset{nodesep=0pt, shortput=nab, npos=0.5, labelsep=2pt, arrowsize=0.2}

      \cvnode{3}{2.8}{0.72}{a}{$a$}
      \cvnodefill{1.5}{1.2}{0.72}{b}{$b$}
      \cvnodefill{4.5}{1.2}{0.72}{c}{$c$}
      \ncnode{0}{0}{0.4}{d}{$d$}
      \ncnode{6}{0}{0.4}{e}{$e$}
      \ncnode{1.5}{-1.2}{0.4}{f}{$f$}
      \ncnode{4.5}{-1.2}{0.4}{g}{$g$}
      \ncnode{6}{-2.8}{0.4}{h}{$h$}

      \ncline[linestyle=dashed, linewidth=0.16cm, linecolor=black]{a}{b}_{$1$}
      \ncline[linewidth=0.16cm, linecolor=black]{a}{c}^{$2$}
      \ncline[linestyle=dashed, linewidth=0.16cm, linecolor=black]{b}{d}_{$1$}
      \psset{npos=0.7}
      \ncline[linestyle=dashed, linewidth=0.16cm, linecolor=black]{b}{f}^{$3$}
      \psset{npos=0.3}
      \ncline[linewidth=0.16cm, linecolor=black]{c}{d}^{$4$}
      \psset{npos=0.5}
      \ncline[linewidth=0.16cm, linecolor=black]{c}{e}^{$2$}
      \ncline[linewidth=0.16cm, linecolor=black]{c}{g}_{$4$}
      \ncline[linestyle=dashed, linewidth=0.16cm, linecolor=black]{d}{f}_{$3$}
      \ncline[linewidth=0.16cm, linecolor=black]{e}{h}^{$2$}
      \ncline[linewidth=0.16cm, linecolor=black]{g}{h}_{$4$}
    \end{pspicture}
    }

    \hspace*{2cm}
    \subfigure[Each node can send messages to all the others.]{
    \begin{pspicture}(-0.5, -3.5)(6.5, 3.5)
      \psset{nodesep=0pt, shortput=nab, npos=0.5, labelsep=2pt, arrowsize=0.2}

      \cvnodefill{3}{2.8}{0.72}{a}{$a$}
      \cvnodefill{1.5}{1.2}{0.72}{b}{$b$}
      \cvnodefill{4.5}{1.2}{0.72}{c}{$c$}
      \ncnode{0}{0}{0.4}{d}{$d$}
      \ncnode{6}{0}{0.4}{e}{$e$}
      \ncnode{1.5}{-1.2}{0.4}{f}{$f$}
      \ncnode{4.5}{-1.2}{0.4}{g}{$g$}
      \ncnode{6}{-2.8}{0.4}{h}{$h$}

      \ncline[linestyle=dashed, linewidth=0.16cm, linecolor=black]{a}{b}_{$1$}
      \ncline[linewidth=0.16cm, linecolor=black]{a}{c}^{$2$}
      \ncline[linestyle=dashed, linewidth=0.16cm, linecolor=black]{b}{d}_{$1$}
      \psset{npos=0.7}
      \ncline[linestyle=dashed, linewidth=0.16cm, linecolor=black]{b}{f}^{$3$}
      \psset{npos=0.3}
      \ncline[linewidth=0.16cm, linecolor=black]{c}{d}^{$4$}
      \psset{npos=0.5}
      \ncline[linewidth=0.16cm, linecolor=black]{c}{e}^{$2$}
      \ncline[linewidth=0.16cm, linecolor=black]{c}{g}_{$4$}
      \ncline[linestyle=dashed, linewidth=0.16cm, linecolor=black]{d}{f}_{$3$}
      \ncline[linewidth=0.16cm, linecolor=black]{e}{h}^{$2$}
      \ncline[linewidth=0.16cm, linecolor=black]{g}{h}_{$4$}
 
    \end{pspicture}
    }
  }
%% If the title of a table is too long to fit in one line,
%% please abbreviate it in a shorter one, and put in [], 
%% which will be displayed in "table of content".  
%% in the text, you may use a long title as shown in {}.
\caption[The dominating set problem in an optical network $N$.]
{The dominating set problem in an optical network $N$.
Some tree decompositions of $N$ are shown in Figure \ref{F:compEx}.}
\label{F:solEx}
\end{center}
\end{figure}

Note that Figure \ref{F:compEx} is cited in the caption of Figure \ref{F:solEx}.

You can explain the meanings of each figure here.

