Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is

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for(int a=0; a<4; ++a)
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for(int b=0; b<4; ++b)
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draw((0,0)--(0,2)--(1,2)--(2,3)--(2,2)--(3,2)--(3,0)--(2,0)--(2,1)--(1,0)--cycle);[/asy]

$ \text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 $
To determine the number of square units enclosed by the given polygon, we can use the Shoelace Theorem (also known as Gauss's area formula for polygons). The Shoelace Theorem states that for a simple polygon with vertices \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\), the area \(A\) is given by:

\[
A = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right|
\]

First, we list the coordinates of the vertices of the polygon in order:
\[
(0,0), (0,2), (1,2), (2,3), (2,2), (3,2), (3,0), (2,0), (2,1), (1,0)
\]

Next, we apply the Shoelace Theorem:

1. Write down the coordinates in a table and repeat the first vertex at the end:

\[
\begin{array}{c|c}
x & y \\
\hline
0 & 0 \\
0 & 2 \\
1 & 2 \\
2 & 3 \\
2 & 2 \\
3 & 2 \\
3 & 0 \\
2 & 0 \\
2 & 1 \\
1 & 0 \\
0 & 0 \\
\end{array}
\]

2. Compute the sum of the products of the coordinates as per the Shoelace Theorem:

\[
\sum (x_i y_{i+1}) = (0 \cdot 2) + (0 \cdot 2) + (1 \cdot 3) + (2 \cdot 2) + (2 \cdot 2) + (3 \cdot 0) + (3 \cdot 0) + (2 \cdot 1) + (2 \cdot 0) + (1 \cdot 0) = 0 + 0 + 3 + 4 + 4 + 0 + 0 + 2 + 0 + 0 = 13
\]

\[
\sum (y_i x_{i+1}) = (0 \cdot 0) + (2 \cdot 1) + (2 \cdot 2) + (3 \cdot 2) + (2 \cdot 3) + (2 \cdot 3) + (0 \cdot 2) + (0 \cdot 2) + (1 \cdot 1) + (0 \cdot 0) = 0 + 2 + 4 + 6 + 6 + 6 + 0 + 0 + 1 + 0 = 25
\]

3. Calculate the area using the Shoelace formula:

\[
A = \frac{1}{2} \left| 13 - 25 \right| = \frac{1}{2} \left| -12 \right| = \frac{1}{2} \times 12 = 6
\]

Thus, the area enclosed by the polygon is \(6\) square units.

The final answer is \(\boxed{6}\).