463 research outputs found
A Constructive Characterisation of Circuits in the Simple (2,2)-sparsity Matroid
We 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 and
the number of edges induced by any is at most 2|X|-2.
Insisting on simplicity results in the Henneberg operation being enough 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.Comment: 22 pages, 6 figures. Changes to presentatio
Graphs of gonality three
In 2013, Chan classified all metric hyperelliptic graphs, proving that
divisorial gonality and geometric gonality are equivalent in the hyperelliptic
case. We show that such a classification extends to combinatorial graphs of
divisorial gonality three, under certain edge- and vertex-connectivity
assumptions. We also give a construction for graphs of divisorial gonality
three, and provide conditions for determining when a graph is not of divisorial
gonality three.Comment: 19 pages, 13 figures; corrected statements of Theorems 1.2 and 4.1,
as well as material in Section
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