Addario-Berry et al. (2008) demonstrated that every even-hole-free graph contains a bisimplicial vertex, which settled a conjecture by Reed.
Conforti et al. (2002) gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in time. da Silva & Vušković (2008) later improved this to . The best currently known algorithm is given by Chang & Lu (2012) and Chang & Lu (2015) which runs in time.
It is unknown whether graph coloring and the maximum independent set problem can be solved in polynomial time on even-hole-free graphs, or whether they are NP-complete. However the maximum clique can be found in even-hole-free graphs in polynomial time.
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- Chang, Hsien-Chih; Lu, Hsueh-I (2012), "A Faster Algorithm to Recognize Even-Hole-Free Graphs", SODA
- Chang, Hsien-Chih; Lu, Hsueh-I (2015), "A Faster Algorithm to Recognize Even-Hole-Free Graphs", Journal of Combinatorial Theory, Series B, 113: 141–161, arXiv:1311.0358, doi:10.1016/j.jctb.2015.02.001
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