Characterizations of matroids with an element lying in a restricted number of circuits
Journal of Combinatorial Optimization(2019)
摘要
matroid M with a distinguished element e_0 ∈ E(M) is a rooted matroid with e_0 being the root. We present a characterization of all connected binary rooted matroids whose root lies in at most three circuits, and a characterization of all connected binary rooted matroids whose root lies in all but at most three circuits. While there exist infinitely many such matroids, the number of serial reductions of such matroids is finite. In particular, we find two finite families of binary matroids ℳ_1 and ℳ_2 and prove the following. (i) For some e_0 ∈ E(M) , M has at most three circuits containing e_0 if and only if the serial reduction of M is isomorphic to a member in ℳ_1 . (ii) If for some e_0 ∈ E(M) , M has at most three circuits not containing e_0 if and only if the serial reduction of M is isomorphic to a member in ℳ_2 . These characterizations will be applied to show that every connected binary matroid M with at least four circuits has a 1-hamiltonian circuit graph.
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关键词
Excluded minor characterizations, Matroid circuit graph, Hamiltonian, 1-hamiltonian
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