Abstract
A classical theorem of Robbins states that the edges of a graph may be oriented, in such a way that an oriented path exists between any source and destination, if and only if the graph is both connected and two-connected (it cannot be disconnected by the removal of an edge). In this paper, an algebraic version of Robbins' result becomes a lemma on Hilbert bases for free abelian groups, which is then applied to generalize his theorem to higher dimensional complexes. An application to cycle bases for graphs is given, and various examples are presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Bjorner,“Homology and shellability," in Matroid Applications, N. White (Ed. ), Cambridge University Press, Cambridge, 1992.
Ken S. Brown, Buildings, Springer-Verlag, New York, 1989.
A. Frank,“Graph connectivity and network flows," a chapter of Handbook of Combinatorics (being prepared).
M. A. Frumkin,“An application of modular arithmetic to the construction of algorithms for solving systems of linear equations," Soviet Mathematics (Doklady), 17 (1976), 1165–1168.
F. R. Giles and W. R. Pulleyblank,“Total dual integrality and integer polyhedra," Linear Algebra and its Applications 25, (1979) 191–196.
J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley Publishing Company, Menlo Park, 1984.
C. hSt. J. A. Nash-Williams," On orientations, connectivity and odd vertex pairings in finite graphs," Canad. J. Math. 12(1960) 555–567.
H. E. Robbins,“A theorem on graphs with an application to a problem of traffic control,” American Math. Monthly 46 (1939), 281–283.
A. Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons, Chichester, 1986.
P. D. Seymour," Sums of circuits," in Graph Theory and Related Topics, J. A. Bondy and U. S. R. Murty (Eds. ), pp. 341–355, Academic Press, New York/Berlin*1979.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Greenberg, P., Loebl, M. Strong Connectivity of Polyhedral Complexes. Journal of Algebraic Combinatorics 5, 117–125 (1996). https://doi.org/10.1023/A:1022413100969
Issue Date:
DOI: https://doi.org/10.1023/A:1022413100969