Twin-width tww(G): minimum d in a contraction sequence reducing G to a single vertex, where at each step, merging two vertices creates at most d 'red' neighbors (disagreeing adjacencies). Bonnet, Kim, Thomassé, Watrigant (2021). Revolutionary parameter.
How to Use
Select graph:
- tww: Twin-width
- Contract: Merge twins
- Red: Mixed entries
Contraction Sequences
Start with G. Repeatedly merge two vertices: combined vertex is adjacent to u if both were, non-adjacent if neither was, or 'red' (mixed). Width = max red degree across all steps. tww = min over all sequences.
Breakthrough Results
tww=0 ⟺ cograph. Bounded tww: all proper minor-closed classes, bounded rw, many more. FPT first-order model checking on bounded tww! Unifies many tractability results. Most exciting parameter since treewidth.
Step-by-Step Instructions
- 1Select graph.
- 2Build contraction.
- 3Track red degree.
- 4Minimize max red.
- 5Compare with cw.