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.^[1] 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.^[2]

Notes

↑ 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 ${\mathcal O}(n^{40})$ ."
↑ 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, doi:10.1016/j.jctb.2015.02.001

External links

"Graphclass: even-cycle-free". Information System on Graph Classes and their Inclusions. External link in |work= (help)

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