  The notion of path-complete $p$-dominance for switching linear systems (in
short, path-dominance) is introduced as a way to generalize the notion of
dominant/slow modes for LTI systems. Path-dominance is characterized by the
contraction property of a set of quadratic cones in the state space. We show
that path-dominant systems have a low-dimensional dominant behavior, and hence
allow for a simplified analysis of their dynamics. An algorithm for deciding
the path-dominance of a given system is presented.
