Abstract
A multi-graph $G$ on n vertices is $(k,l)$-sparse if every subset of $n'≤n$ vertices spans at most $kn'-l$ edges, $0 ≤l < 2k$. $G$ is tight if, in addition, it has exactly $kn - l$ edges. We characterize $(k,l)$-sparse graphs via a family of simple, elegant and efficient algorithms called the $(k,l)$-pebble games. As applications, we use the pebble games for computing components (maximal tight subgraphs) in sparse graphs, to obtain inductive (Henneberg) constructions, and, when $l=k$, edge-disjoint tree decompositions.
Highlights
A graph§ G = (V, E) with n = |V | vertices and m = |E| edges is (k, l)-sparse if every subset of n ≤ n vertices spans at most kn − l edges
Classical results of Nash-Williams [10] and Tutte [16] identify the class of graphs decomposable into k edgedisjoint spanning trees with the tight (k, k)-sparse graphs
The tight (2, 3)-sparse graphs play an important role in Rigidity Theory: they are the generically minimally rigid graphs in dimension 2
Summary
A graph is a (k, a)-arborescence if it can be decomposed into k edge-disjoint spanning trees after the addition of any a edges. There exist many algorithms for decomposing a graph into edge-disjoint trees or forests (some generalizing to and presented in the context of matroid unions), or for recognizing Laman graphs. We describe a family of algorithms, called the (k, l)-pebble games, and prove that they recognize exactly the (k, l)-sparse graphs, 0 ≤ l < 2k.
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