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. Every even-hole-free graph contains a bisimplicial vertex. [citation
last1 = Addario-Berry | first1 = Louigi
last2 = Chudnovsky | first2 = Maria
last3 = Havet | first3 = Frédéric
last4 = Reed | first4 = Bruce
last5 = Seymour | first5 = Paul | author5-link = Paul Seymour
title = Bisimplicial vertices in even-hole-free graphs
journal = Journal of Combinatorial Theory, Series B
year = In press
doi = 10.1016/j.jctb.2007.12.006 ]

Recognition

harvtxt|Conforti|Cornuéjols|Kapoor|Vušković|2002 gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in ${mathcal O}(n^{40})$ time. [harvtxt|Conforti|Cornuéjols|Kapoor|Vušković|2002 present their algorithm and assert that it runs in polynomial time without giving an explicit analysis. harvtxt|Chudnovsky|Kawarabayashi|Seymour|2004 estimate that it runs in "time about ${mathcal O}(n^{40})$ ."] This was later improved to ${mathcal O}(n^{15})$ . [harvtxt|Chudnovsky|Kawarabayashi|Seymour|2004 give an ${mathcal O}(n^{31})$ algorithm and "sketch some improvements that bring the running time down to ${mathcal O}(n^{15})$ ."]

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. [harvtxt|Bienstock|1991]

Notes

References

* citation
last = Bienstock | first = Dan
title = On the complexity of testing for odd holes and induced odd paths
journal = Discrete Mathematics
volume = 90 | issue = 1 | year = 1991 | pages = 85–92
doi = 10.1016/0012-365X(91)90098-M

External links

* cite web
url = http://wwwteo.informatik.uni-rostock.de/isgci/classes/gc_706.html
title = Graphclass: even-cycle-free
work = [http://wwwteo.informatik.uni-rostock.de/isgci/index.html Information System on Graph Class Inclusions]

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