  We study the $\mathcal{L}_p$ induced gain of discrete-time linear switching
systems with graph-constrained switching sequences. We first prove that, for
stable systems in a minimal realization, for every $p \geq 1$, the
$\mathcal{L}_p$-gain is exactly characterized through switching storage
functions. These functions are shown to be the $p$th power of a norm. In order
to consider general systems, we provide an algorithm for computing minimal
realizations. These realizations are \emph{rectangular systems}, with a state
dimension that varies according to the mode of the system. We apply our tools
to the study on the of $\mathcal{L}_2$-gain. We provide algorithms for its
approximation, and provide a converse result for the existence of quadratic
switching storage functions. We finally illustrate the results with a
physically motivated example.
