Fork-decompositions of matroids
In an effort to strengthen the notion of 3-connectivity, the authors study weakened versions of 4-connectivity for matroids. In particular, the authors introduce fork-connectivity, a new type of 4-connectivity for matroids, to allow further progress in matroid representation theory. Conjecture 1.1 (The analogue of Kahn's conjecture). For every finite field \(\text{GF}(q)\), there is an integer \(\nu_q\) such that every vertically 4-connected \(\text{GF}(q)\)-representable matroid has at most \(\nu_q\) inequivalent representations over \(\text{GF}(q)\). The authors believe Conjecture 1.1 to hold for vertically 4-connected matroids and prove that Conjecture 1.1 holds for fork-connected matroids if and only if it holds for vertically 4-connected matroids. The authors prove that every fork-connected matroid can be reduced to a vertically 4-connected matroid by a sequence of segment-cosegment or cosegment-segment exchanges. Since no analogue of Seymour's splitter theorem holds for vertically 4-connected matroids, the authors consider sequentially 4-connected matroids which are closed under duality, contain vertically 4-connected matroids, and for which an analogue of the wheels and whirls theorem has been proven. Fork-connectivity is weaker than sequential 4-connectivity yet maintains many benefits of stronger 4-connectivity notions (e.g. Conjecture 1.1). The authors conjecture that an analogue of Seymour's splitter theorem holds for the class of fork-connected matroids. By taking direct sums and 2-sums of 3-connected forked matroids, the class of forked matroids is formed, which is closed under duality and minors. The authors show this class has finitely many excluded minors.
scientific article; zbMATH DE number 2107251
scientific article