Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present general upper and lower bounds in the setting of general connected orientable matroids, leading to the study of subgraphs of the base graph and the intersection graph of circuits.
We then consider the problem for uniform matroids which is closely related to the notion of (connected) covering numbers in Design Theory. Finally, we also devote special attention to regular matroids as well as some graphic and cographic matroids leading in particular to the topics of (connected) bond and cycle covers in Graph Theory.
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The authors were supported by grant ANR-10-BLAN 0207 and and PICS06316.
The first author was also supported by ANR grant EGOS ANR-12-JS02-002-01 and PEPS grant EROS.
The second author was also supported by the grant LAISLA.
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Knauer, K., Montejano, L.P. & Alfonsín, J.L.R. How Many Circuits Determine an Oriented Matroid?. Combinatorica 38, 861–885 (2018). https://doi.org/10.1007/s00493-016-3556-x
Mathematics Subject Classification (2000)