% Chris Hanusa
% chanusa@qc.cuny.edu	
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\begin{center}\large 
MATH 634, Spring 2014

{\sc Homework 14}

due 5:00{\sc pm} on Wednesday, April 9.
\end{center}

\noindent
{\em Background reading:} {\em Pearls in Graph Theory}, Section 7.1 and 7.2.

\smallskip\noindent
{\color{red} Follow the posted homework guidelines when completing this assignment.}

\smallskip\noindent
Problems {\bf 14D}, {\bf 14E}, and {\bf 14P} should be typed (or written up) and handed in as class starts on Wednesday 4/9. 

\begin{enumerate}
\item[\bf 14D.]  

\begin{itemize}
\item correctness of an algorithm
\item $M$-alternating path
\item $M$-augmenting path
\item stable marriage
\end{itemize}

\item[\bf 14E.] Use the Hungarian algorithm to solve problem 7.2.2. \newline {\bf Important:} Use the initial matching $M=\{(a,4), (c,6), (e,2), (h,5)\}$.


\item[\bf 14P.]
Prove the correctness of the following algorithm to find a spanning tree of a graph. 

{\em [Prove that the algorithm terminates and that the output of the algorithm is a spanning tree of $G$.] }
\begin{itemize}
\item[Input:] A connected graph $G$ with $n$ vertices. 
\item[Preprocess:] Label the vertices $1$ through $n$, color them all white. Let $T$ be a set of edges, initially empty. 
\item[Repeat:] For the lowest numbered white vertex $v$, order the edges incident with $v$. Going through each edge $e$ from first to last, determine if including $e$ in $T$ would create a cycle in $T$. If it would not create a cycle, place $e$ into $T$ ($T$ is growing.) If it would create a cycle, do not add $e$ to $T$; go on to the next edge. Once every incident edge to $v$ has been checked, color $v$ black. If all vertices are black, go on to the next step. Otherwise, repeat this step. 
\item[Output:] Output the graph $T$, a spanning tree of $G$.
\end{itemize}

\end{enumerate}


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