A Note on [k, l]-sparse Graphs

A constructive characterisation of circuits in the simple (2,2)-sparsity matroid

“…The corresponding question for frameworks in the plane was finally settled in 2005 by Jackson and Jordán [10], building upon results of Hendrickson [6], Connelly [2] and most relevantly to this paper, Berg and Jordán [1]. Berg and Jordán's contribution was a recursive characterisation of circuits in M * (2, 3) = M (2,3). Circuits arise because they have the minimum number of edges (as a function of the number of vertices) possible for the realisation to be unique.…”

“…A redundantly rigid framework (G, p) on S 1 × R is a framework such that after deleting any single edge from G the rigidity matroid still has maximal rank. 3,6 illustrates (see also [10, Fig. 6] for the plane case) extending Theorem 1.1 from circuits to 2-connected redundantly rigid graphs is non-trivial.…”

“…(We direct the reader unfamiliar with matroids to [18] for a comprehensive introduction.) There is an elegant recursive construction of the bases (maximal independent sets) in M(k, l) due to Fekete and Szegő [3]. Their result is built on the construction of Tay [20] for k = l. A recursive characterisation of circuits (minimal dependent sets) in M(k, k) can be found as a special case of a theorem of Frank and Szegő [4] on highly k-tree connected multigraphs.…”

A rooted-forest partition with uniform vertex demand

“…Whiteley (1988) proved a generalization of Nash-Williams' theorem by mixing spanning trees and spanning pseudoforests: for two integers k and l with k ≥ l, a graph G = (V , E) can be partitioned into l edge-disjoint spanning trees and k − l spanning pseudoforests if and only if |E| = k|V | − l and |F | ≤ k|V (F )| − l for any non-empty F ⊆ E. Haas (2002) has broadened the range of l: for two integers k and l with k ≤ l ≤ 2k − 1, a graph G = (V , E) satisfies |E| = k|V | − l and |F | ≤ k|V (F )| − l for any non-empty F ⊆ E if and only if it can be partitioned into l edge-disjoint trees such that each vertex is spanned by exactly k of them and any distinct l subtrees (with at least one edge) among them do not span a same vertex subset. Also, Tay (1984), Frank and Szegö (2003), and Fekete and Szegö (2004) provided constructive characterizations of these sparse graphs (i.e., necessary and sufficient conditions to satisfy the counting condition described in terms of inductive graph constructions). Algorithms for checking these counting conditions or computing decompositions were developed in e.g.…”

A constructive characterisation of circuits in the simple (2,1)‐sparse matroid

“…) ≤ 2, so[ 3 ] ( ) ≤ 2. If [ * 3 ] ( ) = 2, then, since has precisely one neighbour in and since is a leaf in [ * 3 − − ], it follows that is a node.…”