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One of the simplest methods of generating a random graph with a given degreesequence is provided by the Monte Carlo Markov Chain method using switches. Theswitch Markov chain converges to the uniform distribution, but generally therate of convergence is not known. After a number of results concerning variousdegree sequences, rapid mixing was established for so-called $P$-stable degreesequences (including that of directed graphs), which covers every previouslyknown 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. Thisfamily has an intimate connection to Tyshkevich-decompositions and strongstability as well.

Half-graphs, other non-stable degree sequences, and the switch Markov chain