2 A tetromino is a two-dimensional shape made by joining four squares edge-to-edge. Joins are along complete edges.\\
(i) Represent each of the tetrominoes below by a graph in which the nodes represent the squares and two nodes are joined by an arc if the squares share a common edge.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{012e87d3-d157-485c-a8bc-2c59c0862f87-2_76_268_1206_420}
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\caption{(A)}
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\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{012e87d3-d157-485c-a8bc-2c59c0862f87-2_145_208_1201_836}
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\caption{(B)}
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\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{012e87d3-d157-485c-a8bc-2c59c0862f87-2_145_144_1201_1165}
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\caption{(C)}
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\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{012e87d3-d157-485c-a8bc-2c59c0862f87-2_145_209_1201_1439}
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\caption{(D)}
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\end{figure}

(ii) Six simply connected graphs with four nodes are shown below. For each graph, either draw a tetromino that can be represented by the graph, as in part (i), or explain why this is not possible.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{012e87d3-d157-485c-a8bc-2c59c0862f87-2_122_126_1603_299}
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\caption{(1)}
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\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{012e87d3-d157-485c-a8bc-2c59c0862f87-2_122_209_1603_529}
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\caption{(2)}
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\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{012e87d3-d157-485c-a8bc-2c59c0862f87-2_122_125_1603_840}
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\caption{(3)}
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\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{012e87d3-d157-485c-a8bc-2c59c0862f87-2_122_127_1603_1082}
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\caption{(4)}
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\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{012e87d3-d157-485c-a8bc-2c59c0862f87-2_122_122_1603_1343}
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\caption{(5)}
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\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{012e87d3-d157-485c-a8bc-2c59c0862f87-2_122_122_1603_1603}
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\caption{(6)}
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\end{figure}

Two tetrominoes are regarded as being the same if one can be rotated or reflected to form the other. Derek claims that each tetromino corresponds to a unique tree with four nodes, and each tree with four nodes corresponds to a unique tetromino. Derek's claim is wrong.\\
(iii) From the diagrams above, find:
\begin{enumerate}[label=(\alph*)]
\item a tetromino whose graph does not correspond to a tree;
\item two different tetrominoes whose graphs correspond to the same tree.

A pentomino is a two-dimensional shape made by joining five squares edge-to-edge. Joins are along complete edges. Two pentominoes are regarded as being the same if one can be rotated or reflected to form the other. There are twelve distinct pentominoes.\\
(iv) When the pentominoes are represented by graphs, as in part (i), there are only four distinct graphs. Draw these four graphs.
\end{enumerate}

\hfill \mbox{\textit{OCR D1 2013 Q2 [10]}}