Define an equivalence relation ~ on the ground set of M by e ~ f if e = f or $\{e,f\}$ is contained in a circuit. The equivalence classes under ~ are the connected components of M. A matroid is the direct sum of its connected components.
i1 : M = matroid graph({{0,1},{0,2},{1,2},{3,4},{4,5}})
o1 = a matroid of rank 4 on 5 elements
o1 : Matroid
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i2 : C = components M
o2 = {a matroid of rank 1 on 1 elements, a matroid of rank 1 on 1 elements, a
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matroid of rank 2 on 3 elements}
o2 : List
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i3 : areIsomorphic(M, fold(C, (a, b) -> a ++ b)) o3 = true |
i4 : G = graph({{0,1},{0,2},{0,3},{0,4},{1,2},{3,4}})
o4 = Graph{0 => {1, 2, 3, 4}}
1 => {0, 2}
2 => {0, 1}
3 => {0, 4}
4 => {0, 3}
o4 : Graph
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i5 : isConnected G o5 = true |
i6 : components matroid G
o6 = {a matroid of rank 2 on 3 elements, a matroid of rank 2 on 3 elements}
o6 : List
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As the examples above show, the connected components of the graphic matroid M(G) need not be the same as the connected components of the graph G (indeed, for any graph G, there exists a connected graph H with M(G) isomorphic to M(H)).