TY - JOUR
T1 - The Holt-Klee condition for oriented matroids
AU - Fukuda, Komei
AU - Moriyama, Sonoko
AU - Okamoto, Yoshio
N1 - Funding Information:
The first author’s research was partially supported by the Swiss National Science Foundation Project 200021-105202.
Funding Information:
The second and third author’s research were partially supported by Grant-in-Aid for Scientific Research from Ministry of Education, Science and Culture, Japan, and Japan Society for the Promotion of Science.
PY - 2009/11
Y1 - 2009/11
N2 - Holt and Klee have recently shown that every (generic) LP orientation of the graph of a d-polytope satisfies a directed version of the d-connectivity property, i.e. there are d internally disjoint directed paths from a unique source to a unique sink. We introduce two new classes HK and HK* of oriented matroids (OMs) by enforcing this property and its dual interpretation in terms of line shellings, respectively. Both classes contain all representable OMs by the Holt-Klee theorem. While we give a construction of an infinite family of non-HK* OMs, it is not clear whether there exists any non-HK OM. This leads to a fundamental question as to whether the Holt-Klee theorem can be proven combinatorially by using the OM axioms only. Finally, we give the complete classification of OM(4, 8), the OMs of rank 4 on 8-element ground set with respect to the HK, HK*, Euclidean and Shannon properties. Our classification shows that there exists no non-HK OM in this class.
AB - Holt and Klee have recently shown that every (generic) LP orientation of the graph of a d-polytope satisfies a directed version of the d-connectivity property, i.e. there are d internally disjoint directed paths from a unique source to a unique sink. We introduce two new classes HK and HK* of oriented matroids (OMs) by enforcing this property and its dual interpretation in terms of line shellings, respectively. Both classes contain all representable OMs by the Holt-Klee theorem. While we give a construction of an infinite family of non-HK* OMs, it is not clear whether there exists any non-HK OM. This leads to a fundamental question as to whether the Holt-Klee theorem can be proven combinatorially by using the OM axioms only. Finally, we give the complete classification of OM(4, 8), the OMs of rank 4 on 8-element ground set with respect to the HK, HK*, Euclidean and Shannon properties. Our classification shows that there exists no non-HK OM in this class.
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U2 - 10.1016/j.ejc.2008.12.012
DO - 10.1016/j.ejc.2008.12.012
M3 - Article
AN - SCOPUS:70349320492
VL - 30
SP - 1854
EP - 1867
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
SN - 0195-6698
IS - 8
ER -