# Even-hole-free graph

In the mathematical area of graph theory, a graph is even-hole-free if it contains no induced cycle with an even number of vertices.

Addario-Berry et al. (2008) demonstrated that every even-hole-free graph contains a bisimplicial vertex, which settled a conjecture by Reed.

## Recognition

Conforti et al. (2002) gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in ${\mathcal {O}}(n^{40})$ time. da Silva & Vušković (2008) later improved this to ${\mathcal {O}}(n^{19})$ . The best currently known algorithm is given by Chang & Lu (2012) and Chang & Lu (2015) which runs in ${\mathcal {O}}(n^{11})$ time.

While even-hole-free graphs can be recognized in polynomial time, it is NP-complete to determine whether a graph contains an even hole that includes a specific vertex.

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.