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 C21C_2^1-Splines II: Combinatorial Characterization

We showed in the first paper of this series that the generic $C_2^1$-cofactor matroid is the unique maximal abstract $3$-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 $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph ${G \in \mathcal{G}}$ there is a graph $H$ with ${\text{tw}(H) \leq c}$ such that $G$ is isomorphic to a subgraph of ${H \boxtimes K_{f(\text{tw}(G))}}$. 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 $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for ${t \geq \max\{s,3\}}$); and the class of $K_t$-minor-free graphs has underlying treewidth ${t-2}$. 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 $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star

Global rigidity of direction-length frameworks

Non UBCUnreviewedAuthor affiliation: Queen Mary LondonGraduat