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 \( {\mathcal O}(n^{19}) \). Chang & Lu (2012) and Chang & Lu (2015) improved this to \({\mathcal O}(n^{11}) \) time. The best currently known algorithm is given by Lai, Lu & Thorup (2020) which runs in \( {\displaystyle {\mathcal {O}}(n^{9})} \) 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
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)
Vušković (2010).
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
Lai, Kai-Yuan; Lu, Hsueh-I; Thorup, Mikkel (2020), "Three-in-a-Tree in Near Linear Time", STOC: 1279–1292, arXiv:1909.07446, doi:10.1145/3357713.3384235
External links
"Even-hole-free graphs", Information System on Graph Classes and their Inclusions
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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