4 research outputs found
A characterization of the base-matroids of a graphic matroid
Let be a matroid on a set and one of its bases. A closed set is saturated with respect to when , where is the rank of . The collection of subsets of such that for every closed saturated set turns out to be the family of independent sets of a new matroid on , called base-matroid and denoted by . In this paper we prove that a graphic matroid , isomorphic to a cycle matroid , is isomorphic to , for every base of , if and only if is direct sum of uniform graphic matroids or, in equivalent way, if and only if is disjoint union of cacti. Moreover we characterize simple binary matroids isomorphic to , with respect to an assigned base
A characterization of the base-matroids of a graphic matroid
Let be a matroid on a set and one of its bases. A closed set is saturated with respect to when , where is the rank of . The collection of subsets of such that for every closed saturated set turns out to be the family of independent sets of a new matroid on , called base-matroid and denoted by . In this paper we prove that a graphic matroid , isomorphic to a cycle matroid , is isomorphic to , for every base of , if and only if is direct sum of uniform graphic matroids or, in equivalent way, if and only if is disjoint union of cacti. Moreover we characterize simple binary matroids isomorphic to , with respect to an assigned base
A characterization of the base-matroids of a graphic matroid
Let be a matroid on a set and one of its bases. A closed set is saturated with respect to when , where is the rank of . The collection of subsets of such that for every closed saturated set turns out to be the family of independent sets of a new matroid on , called base-matroid and denoted by . In this paper we prove that a graphic matroid , isomorphic to a cycle matroid , is isomorphic to , for every base of , if and only if is direct sum of uniform graphic matroids or, in equivalent way, if and only if is disjoint union of cacti. Moreover we characterize simple binary matroids isomorphic to , with respect to an assigned base