  One of the simplest methods of generating a random graph with a given degree
sequence is provided by the Monte Carlo Markov Chain method using switches. The
switch Markov chain converges to the uniform distribution, but generally the
rate of convergence is not known. After a number of results concerning various
degree sequences, rapid mixing was established for so-called $P$-stable degree
sequences (including that of directed graphs), which covers every previously
known rapidly mixing region of degree sequences.
  In this paper we give a non-trivial family of degree sequences that are not
$P$-stable and the switch Markov chain is still rapidly mixing on them. This
family has an intimate connection to Tyshkevich-decompositions and strong
stability as well.
