The “abalone shell” is the simplest contractible, non-collapsible two-complex embedding in three-space.  This is because it has a single “vertex”. For a picture, see Figure 3 of Matveev’s [article][1].

By the way, Bing’s house has two vertices, not four (as suggested by your figure). See Figure 4 of Matveev’s article.


  [1]: https://archive.mpim-bonn.mpg.de/id/eprint/2389/2/preprint_1998_67.pdf