Almost balanced biased graph representations of frame matroids

Abstract

Given a 3-connected biased graph Ω with a balancing vertex, and with frame matroid F (Ω) nongraphic and 3-connected, we determine all biased graphs Ω with F (Ω ) = F (Ω). As a consequence, we show that if M is a 4connected nongraphic frame matroid represented by a biased graph Ω having a balancing vertex, then Ω essentially uniquely represents M . More precisely, all biased graphs representing M are obtained from Ω by replacing a subset of the edges incident to its unique balancing vertex with unbalanced loops.

The $9$-connected Excluded Minors for the Class of Quasi-graphic Matroids

“…Assume that G has a blocking vertex u. Since each circuit in U 4,6 has five elements and 6 for each edge f ∈ st G (u), there are exactly two edges joining u and v i for each…”

“…We call the partition given by the equivalence classes of ∼ v the standard partition of st H (v). For more details, the reader can refer to ([5], Section 2) or ( [6], Section 1). Definitions and results introduced in this paragraph will be only used in the proof of Lemma 5.8.…”

Matrix representations of frame and lifted-graphic matroids correspond to gain functions

“…Let J be the set of joints of (G, B) that are not incident to u, and denote by δ(u) the set of links incident to u. It is not difficult to check that the theta property implies that for each pair e, e ∈ δ(u), either all cycles containing both e and e are balanced or all cycles containing both e and e are unbalanced (for details, see [5,Section 1]). Observe that just as for a pair e, e ∈ δ(u) for which every cycle containing both e and e is balanced, for every pair f, f ∈ J, every path linking the endpoint of f with the endpoint of f together with…”

“…Thus we just need consider almost-balanced and properly unbalanced biased graphs. The collection of unbalanced cycles in almost-balanced biased graphs is highly structured and well-understood (see [5]). For each almost-balanced biased graph (G, B) there is a family of almost-balanced biased graphs R (G,B) , each of which represents the frame matroid F (G, B).…”

“…Then for each pair of edges e 1 , e 2 ∈ Σ G (u), either every minimal path linking the endpoints of e 1 and e 2 in G − u together with {e 1 , e 2 } forms a circuit in F (G, B), or all such paths together with {e 1 , e 2 } are independent in F (G, B). This defines an equivalence relation on Σ G (u)[5, Lemma 1.4]. We call these equivalence classes the unbalancing classes of Σ G (u).…”