\documentstyle[11pt]{article}
\pagestyle{empty}
\textwidth=6.5in
\textheight=9in
\evensidemargin=0in
\oddsidemargin=0in
\topmargin=-0.3in
\topskip=0pt
\baselineskip=12pt
\parskip=0pt
\parindent=1em
\newtheorem{obs}{Observation}
\newtheorem{defn}{Definition}
\newtheorem{thm}{Theorem}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\begin{document}
\Large
   \begin{figure}
        \begin{center}
\begin{eqnarray*}
{\cal X} & = & \{x_{1}, x_{2}, \ldots, x_{3q} \} \\
{\cal C} & = & \{ \{x_{1}, x_{2}, x_{4} \}, \{x_{3}, x_{4}, x_{5}\}, 
	\ldots \} \\
{\cal Q} & & Does \ C \ contain \ an \ exact \ cover \ C' \ of \ {\cal X} ?
\end{eqnarray*}
        \end{center}

\vspace{1in}
\large
	\begin{center}
        \begin{picture}(220,170)(0,-90)

                \put(30,0){\circle*{10}}
                \put(30,5){\line(0,1){10}}
                \put(30,20){\circle{10}}
                \put(30,25){\line(1,1){11}}
                \put(30,25){\line(-1,1){11}}
                \put(45,40){\circle{10}}
                \put(15,40){\circle{10}}
                \put(13,44){\line(-1,1){11}}
                \put(0,60){\circle*{10}}
                \put(0,65){\line(0,1){10}}
                \put(0,80){\circle*{10}}
                \put(11,78) {\makebox(0,0)[b]{$a_{1}$}}
                \put(11,58) {\makebox(0,0)[b]{$b_{1}$}}
                \put(5,38){\makebox(0,0)[b]{$f_{1}$}}
                \put(45,49){\makebox(0,0)[b]{$e_{1}$}}
                \put(41,18){\makebox(0,0)[b]{$d_{1}$}}
                \put(41,-2){\makebox(0,0)[b]{$c_{1}$}}

                \put(30,-5){\line(0,-1){40}}
                %\put(30,-5){\line(-1,-2){21}}
                \put(5,-45){\line(1,2){22}}
                %\put(30,-5){\line(4,-3){54}}
                \put(85,-45){\line(-4,3){54}}

                \put(85,0){\circle{10}}
                \put(85,5){\line(0,1){10}}
                \put(85,20){\circle*{10}}
                \put(85,25){\line(1,1){11}}
                \put(85,25){\line(-1,1){11}}
                \put(100,40){\circle*{10}}
                \put(70,40){\circle{10}}
                \put(68,44){\line(-1,1){11}}
                \put(55,60){\circle{10}}
                \put(55,65){\line(0,1){10}}
                \put(55,80){\circle*{10}}
                \put(66,78) {\makebox(0,0)[b]{$a_{2}$}}
                \put(66,58) {\makebox(0,0)[b]{$b_{2}$}}
                \put(60,38){\makebox(0,0)[b]{$f_{2}$}}
                \put(100,49){\makebox(0,0)[b]{$e_{2}$}}
                \put(96,18){\makebox(0,0)[b]{$d_{2}$}}
                \put(96,-2){\makebox(0,0)[b]{$c_{2}$}}

                \put(85,-5){\line(0,-1){40}}
                %\put(85,-5){\line(-1,-2){22}}
                \put(57.5,-45){\line(1,2){22}}
                %\put(85,-5){\line(1,-2){22}}
                \put(112.5,-45){\line(-1,2){22}}

                \put(140,20){\makebox(0,0)[b]{\ldots}}

                \put(200,0){\circle*{10}}
                \put(200,5){\line(0,1){10}}
                \put(200,20){\circle{10}}
                \put(200,25){\line(1,1){11}}
                \put(200,25){\line(-1,1){11}}
                \put(215,40){\circle{10}}
                \put(185,40){\circle{10}}
                \put(183,44){\line(-1,1){11}}
                \put(170,60){\circle*{10}}
                \put(170,65){\line(0,1){10}}
                \put(170,80){\circle*{10}}
                \put(181,78) {\makebox(0,0)[b]{$a_{m}$}}
                \put(181,58) {\makebox(0,0)[b]{$b_{m}$}}
                \put(175,38){\makebox(0,0)[b]{$f_{m}$}}
                \put(215,49){\makebox(0,0)[b]{$e_{m}$}}
                \put(212,18){\makebox(0,0)[b]{$d_{m}$}}
                \put(212,-2){\makebox(0,0)[b]{$c_{m}$}}



                \put(5,-50){\circle{10}}
                \put(5,-65){\line(0,1){10}}
                \put(5,-70){\circle{10}}
                \put(17,-50){\makebox(0,0)[b]{$x_{1}$}}
                \put(17,-70){\makebox(0,0)[b]{$y_{1}$}}

                \put(30,-50){\circle{10}}
                \put(30,-65){\line(0,1){10}}
                \put(30,-70){\circle{10}}
                \put(42,-50){\makebox(0,0)[b]{$x_{2}$}}
                \put(42,-70){\makebox(0,0)[b]{$y_{2}$}}

                \put(57.5,-50){\circle{10}}
                \put(57.5,-65){\line(0,1){10}}
                \put(57.5,-70){\circle{10}}
                \put(69.5,-50){\makebox(0,0)[b]{$x_{3}$}}
                \put(69.5,-70){\makebox(0,0)[b]{$y_{3}$}}

                \put(85,-50){\circle{10}}
                \put(85,-65){\line(0,1){10}}
                \put(85,-70){\circle{10}}
                \put(97,-50){\makebox(0,0)[b]{$x_{4}$}}
                \put(97,-70){\makebox(0,0)[b]{$y_{4}$}}

                \put(112.5,-50){\circle{10}}
                \put(112.5,-65){\line(0,1){10}}
                \put(112.5,-70){\circle{10}}
                \put(124.5,-50){\makebox(0,0)[b]{$x_{5}$}}
                \put(124.5,-70){\makebox(0,0)[b]{$y_{5}$}}

                \put(150,-70){\makebox(0,0)[b]{\ldots}}
                \put(205,-50){\circle{10}}
                \put(205,-65){\line(0,1){10}}
                \put(205,-70){\circle{10}}
                \put(217,-50){\makebox(0,0)[b]{$x_{3q}$}}
                \put(217,-70){\makebox(0,0)[b]{$y_{3q}$}}


        \end{picture}
	\end{center}
        \begin{center}
\begin{eqnarray*}
{\cal K} & = & 3m \\
{\cal S} & = &  1-maximal \ \ n_{p}-set \ of \ size \ {\cal K}\\
\end{eqnarray*}
        \end{center}
        \end{figure}

%        \end{center}
\end{document}
