C

In a common emitter configuration, the relationships between the current gains are defined as follows:

1. The current gain in output to input for a transistor is given by:
$$ \alpha = \frac{I_C}{I_E} $$
where $I_C$ is the collector current and $I_E$ is the emitter current.

2. The current gain defined in terms of the base current is:
$$ \beta = \frac{I_C}{I_B} $$
where $I_B$ is the base current.

3. The relationship between these two parameters can be expressed as:
$$ \alpha = \frac{\beta}{1 + \beta} $$
Rearranging gives:
$$ \beta = \frac{\alpha}{1 - \alpha} $$

Thus, if we rearrange to find $\alpha$ in terms of $\beta$, we find one of the options presented in the question:

The incorrect relationship is in option (c) $$ \alpha = \frac{\beta^2}{1 + \beta^2} $$
This is not a valid equation that relates $\alpha$ and $\beta$.
Therefore, option (c) is the incorrect relationship.