Count Matroids of Group-Labeled Graphs

Abstract

A graph G = (V, E) is called (k, )-sparse if |F | ≤ k|V (F )| − for any nonempty F ⊆ E, where V (F ) denotes the set of vertices incident to F . It is known that the family of the edge sets of (k, )-sparse subgraphs forms the family of independent sets of a matroid, called the (k, )-count matroid of G. In this paper we shall investigate lifts of the (k, )-count matroids by using group labelings on the edge set. By introducing a new notion called near-balancedness, we shall identify a new class of matroids whose independence condition is described as a count condition of the form |F | ≤ k|V (F )| − + α ψ (F ) for some function α ψ determined by a given group labeling ψ on E.

Pairing Symmetries for Euclidean and Spherical Frameworks

“…(The block matrix corresponding to the trivial representation of Γ is the orbit matrix.) Combinatorial characterisations of Γ-regular infinitesimally rigid frameworks in the plane have been obtained via this approach for a selection of cyclic groups (where the action θ : Γ → Aut(G) is free on the vertex set) [13,14,32]. The problem remains open for all other groups.…”

Rigidity of symmetric frameworks in normed spaces

“…Also note that, in any dimension, the (k, l, m)-gain tight counts given by Corollary 2.19 are the bases of a matroid as was observed in [15]. (Note however that this matroidal property does not hold for arbitrary triples k, l, m ∈ N. Indeed it fails in some rigidity contexts [3]. )…”

“…In Sect. 3, a new inductive construction is obtained for the class of (2, 2, 0)-gain-tight graphs (Theorem 3.16). Previous recursive characterisations of (2, 2, m)-gain-tight graphs, with m ∈ {1, 2}, can be found in [15].…”

Maximal Matroids in Weak Order Posets

“…The concept of count matroids has been extended to hypergraphs [22] and to grouplabeled graphs [13]. The technique in this section can be adapted to both settings.…”