Even-hole-free graph

Short description: Graph containing no induced cycles with an even number of nodes

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. More precisely, the definition may allow the graph to have induced cycles of length four, or may also disallow them: the latter is referred to as even-cycle-free graphs.

(Addario-Berry Chudnovsky) demonstrated that every even-hole-free graph contains a bisimplicial vertex (a vertex whose neighborhood is the union of two cliques), which settled a conjecture by Reed. The proof was later shown to be flawed by (Chudnovsky Seymour), who gave a correct proof.

Recognition

(Conforti Cornuéjols) gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in [math]\displaystyle{ {\mathcal O}(n^{40}) }[/math] time. (da Silva Vušković) later improved this to [math]\displaystyle{ {\mathcal O}(n^{19}) }[/math]. (Chang Lu) and (Chang Lu) improved this to [math]\displaystyle{ {\mathcal O}(n^{11}) }[/math] time. The best currently known algorithm is given by (Lai Lu) which runs in [math]\displaystyle{ {\mathcal O}(n^9) }[/math] 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.

References

• Addario-Berry, Louigi (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 
• "Detecting even holes", Journal of Graph Theory 48 (2): 85–111, 2004, doi:10.1002/jgt.20040 
• Conforti, Michele (January 2002a), "Even-hole-free graphs part I: Decomposition theorem", Journal of Graph Theory 39 (1): 6–49, doi:10.1002/jgt.10006, http://eprints.whiterose.ac.uk/74361/2/decomp-even.pdf 
• Conforti, Michele (August 2002b), "Even-hole-free graphs part II: Recognition algorithm", Journal of Graph Theory 40 (4): 238–266, doi:10.1002/jgt.10045, http://eprints.whiterose.ac.uk/74360/2/algo-even.pdf 
• da Silva, Murilo V.G. (2008), Decomposition of even-hole-free graphs with star cutsets and 2-joins, http://www.comp.leeds.ac.uk/vuskovi/star.ps 
• Chang, Hsien-Chih; Lu, Hsueh-I (January 2012), "A Faster Algorithm to Recognize Even-Hole-Free Graphs", SODA '12: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms: 1286–1297, doi:10.1137/1.9781611973099.101, ISBN 978-1-61197-210-8 
• Chang, Hsien-Chih; Lu, Hsueh-I (July 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 
• "Even-hole-free graphs: a survey", Applicable Analysis and Discrete Mathematics 4 (2): 219–240, 2010, doi:10.2298/AADM100812027V, http://eprints.whiterose.ac.uk/74347/2/ehfsurvey.pdf 
• Lai, Kai-Yuan; Lu, Hsueh-I (2020), "Three-in-a-tree in near linear time", in Makarychev, Konstantin; Makarychev, Yury; Tulsiani, Madhur et al., Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22–26, 2020, Association for Computing Machinery, pp. 1279–1292, doi:10.1145/3357713.3384235 
• Chudnovsky, Maria; Seymour, Paul (2023), "Even-hole-free graphs still have bisimplicial vertices", Journal of Combinatorial Theory, Series B, doi:10.1016/j.jctb.2023.02.009, https://www.sciencedirect.com/science/article/pii/S0095895623000151 

External links

• "Even-hole-free graphs", Information System on Graph Classes and their Inclusions, http://www.graphclasses.org/classes/gc_547.html