A matroid view of key theorems for edge-swapping algorithms

We demonstrate that two key theorems of Amaldi, Liberti, Maffiolo and Maculan (2009), which they presented with rather complicated proofs, can be more easily and cleanly established using a simple and classical property of binary matroids. Besides a simpler proof, we see that both of these key results are manifestations of the same essential property. Our goal is to demonstrate that two graph-theoretic theorems from [1] are direct manifestations of a simple and classical property of binary matroids. This follows the line of matroids sometimes serving as a means to simplify and unify combinatorial theorems involving graphs and coordinatized vector spaces. We assume some very basic familiarity with matroid terminology and theory (see [2]), and we rely on elementary notions from graph theory. For a matroid M , we denote its dual matroid by M∗ (i.e., the matroid on the same ground set as M but having its set of bases to be the set of complements of bases of M). For S ⊂ E, we let S denote the complement of S in E. For S,U ⊂ E, we denote the symmetric difference (S \ U) ∪ (U \ S) by S∆U . Let G be a finite connected graph with edge set E. Let T ⊂ E be a spanning tree of G. For an edge f 6∈ T , let C(T, f) be the unique cycle of G contained in T ∪{f}. Similarly, for an edge e ∈ T , let D(T, e) be the unique cocycle of G contained in T ∪{e}. From the point of view of matroid theory, T is a base of the graphic matroid of G and T c is a base of the cographic matroid of G, so C(T, f) and D(T, e) are fundamental circuits (in the matroid sense). For an edge f 6∈ T and an edge e ∈ C(T, f) \ {f}, T ∪ {f} \ {e} is also a spanning tree of G, and hence a base of the graphic matroid of G. Of course then, (T ∪{f}\{e}) is also the complement of a spanning tree of G, so (T ∪ {f} \ {e}) is a base of the cographic matroid of G. With our notation, Theorem 1 of [1] is as follows Theorem 1. Let e ∈ T , f ∈ T , e 6= f , g ∈ C(T, e) ∩ C(T, f). Then C(T ∪ {e} \ {g}, f) = C(T, e) ∆ C(T, f). The proof of Theorem 1 presented in [1] is not egregiously lengthy, but it is a bit ad hoc. However, the authors also present a Theorem 9, which they demonstrate with a long (three page) proof that is relegated to an appendix. In the way that it is stated, it is not immediately clear that the result is dual to Theorem 1. Here, we state it in a way that makes it easier to directly compare. Theorem 9. Let e ∈ T , f ∈ T , e 6= f , g / ∈ D(T, e) ∩D(T, f). Then D(T ∪ {e} \ {g}, f) = D(T, e) ∆ D(T, f). Date: January 20, 2012. Revised March 27, 2012.