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\begin{document}

\centerline{\bf HW 10 CMSC 452}
\centerline{\bf Morally Due TUES April 15 11:00AM} 
\centerline{\bf Dead-Cat Due THU April 17 11:00AM} 


\begin{enumerate}

\item
(50 points) 

{ \it Def} Let $G=(V,E)$ be a graph. 
A {\it vertex cover for $G$ of size $k$} is a set $U\subseteq V$
such that 
\begin{itemize} 
\item 
$|U|=k$, and
\item
For every $(a,b)\in E$ either $a\in U$ or $b\in U$ (or both).
\end{itemize} 

\smallskip
$\VC = \{ (G,k) \colon \hbox{$G$ has a Vertex Cover of size $\le k$} \}.$

\smallskip
It is known that $\VC$ is NP-complete.

\begin{enumerate}
\item
(10 points)
Give a connected graph $G$ on 1000 vertices that has a vertex cover of size 1.
\item
(10 points)
Give a connected graph $G$ on 1000 vertices so that the smallest vertex cover for it has size 999.
\item
(10 points)
Give a connected graph $G$ on 1000 vertices so that the following are true.
\begin{itemize}
\item
$G$ has a vertex cover of size 500.
\item
$G$ does not have a vertex cover of size 499.
\end{itemize}
\item
(20 points) 
Let 

$\VC_{1000} = \{ G \colon \hbox{$G$ has a Vertex Cover of size $1000$} \}.$

Show that $\VC_{1000}\in\P$. 

\item
(0 points)
Think about: Your algorithm in Part d ran in time $O(n^d)$ for some $d$.
Do you think there is an algorithm with a substantially lower value of $d$? 

\item 
(0 points) 
Think about but do not hand in: 
Consider the greedy algorithm to find a Vertex Cover: 
Remove the vertex of highest degree put it in set $X$.
When you remove the vertex also remove the edges attached to it
(obviously). 
Repeat: remove the vertex of highest degree put it in set $X$.
Keep doing this until the graph no longer has any edges.
$X$ is a vertex cover. 

Find a graph where this algorithm DOES NOT give the min sized vertex cover.
\end{enumerate}

\newpage

\item
(50 points) 

{ \it Def} Let $G=(V,E)$ be a graph. 
A {\it dominating set for $G$ of size $k$} is a set $D\subseteq V$
such that 
\begin{itemize} 
\item 
$|D|=k$, and
\item
For every $v\in V$ either $v\in D$ or some neighbor of $v$ is in $D$.
\end{itemize} 

\smallskip
$\DS = \{ (G,k) \colon \hbox{$G$ has a Dominating Set of size $k$} \}.$

\smallskip
It is known that $\DS$ is NP-complete.

\begin{enumerate}
\item
(10 points)
Give a connected graph $G$ on 1000 vertices that has a dom set of size 1.
\item
(10 points)
Give a graph $G$ on 1000 vertices so that the smallest dom set for it has size 1000.
(Hint: It will NOT be connected.) 
\item
(10 points)
Give a graph $G$ on 1000 vertices so that the following are true.
\begin{itemize}
\item
$G$ has a dom set of size 500.
\item
$G$ does not have a dom set of size 499.
\end{itemize}
(Hint: It will NOT be connected.) 
\item
(20 points) 
Let 

$\DS_{1000} = \{ G \colon \hbox{$G$ has a Dominating Set of size $1000$} \}.$

Show that $\DS_{1000}\in\P$.

\item
(0 points)
Think about: Your algorithm in Part d ran in time $O(n^d)$ for some $d$.
Do you think there is an algorithm with a substantially lower value of $d$? 


\item
(0 points) 
Think about but do not hand in: 
Consider the greedy algorithm to find a Dom Set: 
Remove the vertex of highest degree put it in set $X$.
When you remove the vertex also remove the edges attached to it
(obviously) AND the vertices on those edges. 
Repeat: remove the vertex of highest degree put it in set $X$.
Keep doing this until the graph no longer has any edges.
$X$ is a dom set. 

Find a graph where this algorithm  DOES NOT give the min sized dom set.

\end{enumerate}

\end{enumerate}

\end{document}