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 ${\displaystyle {\mathcal {O}}(n^{40})}$ time.[1] da Silva & Vušković (2008) later improved this to ${\displaystyle {\mathcal {O}}(n^{19})}$. The best currently known algorithm is given by Chang & Lu (2012) and Chang & Lu (2015) which runs in ${\displaystyle {\mathcal {O}}(n^{11})}$ 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.[2]

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.[3]

Notes

1. Conforti et al. (2002) present their algorithm and assert that it runs in polynomial time without giving an explicit analysis. Chudnovsky, Kawarabayashi & Seymour (2004) estimate that it runs in "time about ${\displaystyle {\mathcal {O}}(n^{40})}$."
2. Bienstock (1991)

References

• Addario-Berry, Louigi; Chudnovsky, Maria; Havet, Frédéric; Reed, Bruce; Seymour, Paul (2008), "Bisimplicial vertices in even-hole-free graphs", Journal of Combinatorial Theory, Series B, 98 (6): 1119–1164, doi:10.1016/j.jctb.2007.12.006
• Bienstock, Dan (1991), "On the complexity of testing for odd holes and induced odd paths", Discrete Mathematics, 90 (1): 85–92, doi:10.1016/0012-365X(91)90098-M
• Chudnovsky, Maria; Kawarabayashi, Ken-ichi; Seymour, Paul (2004), "Detecting even holes", Journal of Graph Theory, 48 (2): 85–111, doi:10.1002/jgt.20040
• Conforti, Michele; Cornuéjols, Gérard; Kapoor, Ajai; Vušković, Kristina (2002), "Even-hole-free graphs part I: Decomposition theorem", Journal of Graph Theory, 39 (1): 6–49, doi:10.1002/jgt.10006
• Conforti, Michele; Cornuéjols, Gérard; Kapoor, Ajai; Vušković, Kristina (2002), "Even-hole-free graphs part II: Recognition algorithm", Journal of Graph Theory, 40 (4): 238–266, doi:10.1002/jgt.10045
• da Silva, Murilo V.G.; Vušković, Kristina (2008), Decomposition of even-hole-free graphs with star cutsets and 2-joins
• 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
• Vušković, Kristina (2010), "Even-hole-free graphs: a survey", Applicable Analysis and Discrete Mathematics, 4 (2): 219–240, doi:10.2298/AADM100812027V, JSTOR 43666110, MR 2724633