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

Abstract

a b s t r a c tWe provide a constructive characterisation of circuits in the simple (2, 2)-sparsity matroid. A circuit is a simple graph G = (V , E) with |E| = 2|V | − 1 where the number of edges induced by any X V is at most 2|X | − 2. Insisting on simplicity results in the Henneberg 2 operation being adequate only when the graph is sufficiently connected. Thus we introduce 3 different join operations to complete the characterisation. Extensions are discussed to when the sparsity matroid is connected and this is applied to the theory of frameworks on surfaces, to provide a conjectured characterisation of when frameworks on an infinite circular cylinder are generically globally rigid.

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

“…We now establish connectivity conditions that guarantee that, in a 3-connected, essentially 5-edgeconnected (2, 1)-circuit with ( ) = 3, there exist nodes not contained in copies of 4 . Since (the vertex set of) 4 is semi-critical, this is significantly harder than the corresponding result in [11]. We will first prove two lemmas for (2, 1)-circuits that contain proper critical sets, before proving their (simpler) analogues for (2, 1)-circuits that contain no proper critical sets.…”

“…Now we can prove our main result. In a similar manner to that adopted in [1] and [11] we refer to applications of Lemmas 4.1 to 4.7 to combine two (2, 1) circuits as taking sums of connected components. (⇒) Suppose that ∈ (2, 1).…”

“…A related construction for circuits in the (2, 3)-sparse matroid [1] was a vital aspect of the characterisation of global rigidity in the plane [6] and we expect that the construction here, along with the construction for circuits in the simple (2, 2)-sparse matroid, will be crucial in establishing analogues for global rigidity on surfaces supporting either two (e.g., the cylinder) or one (e.g., the cone or torus) tangentially acting isometries. See [11,Conjecture 5.7] and [7,Conjecture 9.1].…”

Global rigidity of generic frameworks on the cylinder

“…If G is 3-connected then the statement is [18, Theorem 1.2] so we may assume that G is not 3-connected. Since G is an M * 2,2 -circuit, G is 2-connected by [18,Lemma 2.3]. Since G has no nontrivial 2-vertex separation and G = H 1 , every 2-vertex-separation (F i , F j ) of G has F i = K 4 and F j = K 4 .…”

“…[18, Theorem 5.4] Suppose that G is a graph. Then M * 2,2 (G) is connected if and only if G is 2-connected and redundantly rigid on Y.…”