2 A tetromino is a two-dimensional shape made by joining four squares edge-to-edge. Joins are along complete edges.
- 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.
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\caption{(A)}
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\caption{(B)}
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\caption{(C)}
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\caption{(D)}
\end{figure} - 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.
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\caption{(1)}
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\caption{(2)}
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\caption{(3)}
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\caption{(4)}
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\caption{(5)}
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\caption{(6)}
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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. - From the diagrams above, find:
(a) a tetromino whose graph does not correspond to a tree;
(b) 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. - When the pentominoes are represented by graphs, as in part (i), there are only four distinct graphs. Draw these four graphs.