8 research outputs found
Constructing isostatic frameworks for the ℓ1and ℓ∞-plane
We use a new coloured multi-graph constructive method to prove that if the edge-set of a graph G=(V,E) has a partition into two spanning trees T1 and T2 then there is a map p:V→R2, p(v)=(p1(v),p2(v)), such that |pi(u)−pi(v)|⩾|p3−i(u)−p3−i(v)| for every edge uv in Ti(i=1,2). As a consequence, we solve an open problem on the realisability of minimally rigid bar-joint frameworks in the ℓ1 or ℓ∞-plane. We also show how to adapt this technique to incorporate symmetry and indicate several related open problems on rigidity, redundant rigidity and forced symmetric rigidity in normed spaces
Abstract 3-Rigidity and Bivariate -Splines II: Combinatorial Characterization
We showed in the first paper of this series that the generic -cofactor
matroid is the unique maximal abstract -rigidity matroid. In this paper we
obtain a combinatorial characterization of independence in this matroid. This
solves the cofactor counterpart of the combinatorial characterization problem
for the rigidity of generic 3-dimensional bar-joint frameworks. We use our
characterization to verify that the counterparts of conjectures of Dress (on
the rank function) and Lov\'{a}sz and Yemini (which suggested a sufficient
connectivity condition for rigidity) hold for this matroid
Auction algorithm sensitivity for multi-robot task allocation
We consider the problem of finding a low-cost allocation and ordering of
tasks between a team of robots in a d-dimensional, uncertain, landscape, and
the sensitivity of this solution to changes in the cost function. Various
algorithms have been shown to give a 2-approximation to the MinSum allocation
problem. By analysing such an auction algorithm, we obtain intervals on each
cost, such that any fluctuation of the costs within these intervals will result
in the auction algorithm outputting the same solution
Product structure of graph classes with bounded treewidth
We show that many graphs with bounded treewidth can be described as subgraphs
of the strong product of a graph with smaller treewidth and a bounded-size
complete graph. To this end, define the "underlying treewidth" of a graph class
to be the minimum non-negative integer such that, for some
function , for every graph there is a graph with
such that is isomorphic to a subgraph of . We introduce disjointed coverings of graphs
and show they determine the underlying treewidth of any graph class. Using this
result, we prove that the class of planar graphs has underlying treewidth 3;
the class of -minor-free graphs has underlying treewidth (for ); and the class of -minor-free graphs has underlying
treewidth . In general, we prove that a monotone class has bounded
underlying treewidth if and only if it excludes some fixed topological minor.
We also study the underlying treewidth of graph classes defined by an excluded
subgraph or excluded induced subgraph. We show that the class of graphs with no
subgraph has bounded underlying treewidth if and only if every component of
is a subdivided star, and that the class of graphs with no induced
subgraph has bounded underlying treewidth if and only if every component of
is a star
Global rigidity of direction-length frameworks
Non UBCUnreviewedAuthor affiliation: Queen Mary LondonGraduat