Specialization of polynomial covers of prime degree

Let K be a complete field of unequal characteristics $(0,p)$. The aim of this paper is to describe the the semi-stable models for covers $\bold P^1_K@>>>\bold P^1_K$ofdegreep,unramifiedoutside$r\leq p$ points and totally ramified above one of them, under the assumption that the ramified locus has a particular reduction type (which always occurs if $r\le 4$). We are principally concerned with the minimal semi-stable models which separate the ramified fibers.