The Rigidity of Spherical Frameworks: Swapping Blocks and Holes

A significant range of geometric structures whose rigidity is explored for both practical and theoretical purposes are formed by modifying generically isostatic triangulated spheres. In the block and hole structures (P, p), some edges are removed to make holes, and other edges are added to create rigid sub-structures called blocks. Previous work noted a combinatorial analogy in which blocks and holes played equivalent roles. In this paper, we connect stresses in such a structure (P, p) to first-order motions in a swapped structure (P', p), where holes become blocks and blocks become holes. When the initial structure is geometrically isostatic, this shows that the swapped structure is also geometrically isostatic, giving the strongest possible correspondence. We use a projective geometric presentation of the statics and the motions, to make the key underlying correspondences transparent.Comment: 36 pages, 9 figure

3/2 Firefighters are not enough

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.Comment: 8 page

A note on bounds for the cop number using tree decompositions

In this short note, we supply a new upper bound on the cop number in terms of tree decompositions. Our results in some cases extend a previously derived bound on the cop number using treewidth

Hyperopic Cops and Robbers

We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the usual cop number, the hyperopic cop number is at most 3. The hyperopic cop number is considered for countable graphs, and it is shown that for connected chains of graphs, the hyperopic cop density can be any real number in $[0,1/2].

On Packing Colorings of Distance Graphs

The {\em packing chromatic number} $\chi_{\rho}(G)$ of a graph $G$ is the least integer $k$ for which there exists a mapping $f$ from $V(G)$ to $\{1,2,\ldots ,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. This paper studies the packing chromatic number of infinite distance graphs $G(\mathbb{Z},D)$, i.e. graphs with the set $\mathbb{Z}$ of integers as vertex set, with two distinct vertices $i,j\in \mathbb{Z}$ being adjacent if and only if $|i-j|\in D$. We present lower and upper bounds for $\chi_{\rho}(G(\mathbb{Z},D))$, showing that for finite $D$, the packing chromatic number is finite. Our main result concerns distance graphs with $D=\{1,t\}$ for which we prove some upper bounds on their packing chromatic numbers, the smaller ones being for $t\geq 447$: $\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq 40$ if $t$ is odd and $\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq 81$ if $t$ is even

The Firefighter Problem: A Structural Analysis

We consider the complexity of the firefighter problem where b>=1 firefighters are available at each time step. This problem is proved NP-complete even on trees of degree at most three and budget one (Finbow et al.,2007) and on trees of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this paper, we provide further insight into the complexity landscape of the problem by showing that the pathwidth and the maximum degree of the input graph govern its complexity. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any fixed budget b>=1. We then show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter "pathwidth" and "maximum degree" of the input graph

Structures and melting in infinite gold nanowires

The temperature dependence of structural properties for infinitely long gold nanowires is studied. The molecular dynamics simulation method and the embedded-atom potential are used. The wires constructed at T=0 K with a face-centered cubic structure and oriented along the (111), (110), and (100) directions are investigated. It was found that multiwalled structures form in all these nanowires. The coaxial cylindrical shells are the most pronounced and well-formed for an initial fcc(111) orientation. The shells stabilize with increasing temperature above 300 K. All nanowires melt at T<1100 K, i.e., well below the bulk melting temperature.Comment: 8 pages, 3 jpg and 2 ps figure

Fire Containment in Planar Graphs

In a graph $G$, a fire starts at some vertex. At every time step, firefighters can protect up to $k$ vertices, and then the fire spreads to all unprotected neighbours. The $k$-surviving rate $\rho_k(G)$ of $G$ is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs \cG we are interested in the minimum value $k$ such that $\rho_k(G)\ge\epsilon$ for some constant $\epsilon>0$ and all G\in\cG i.e., such that linearly many vertices are expected to be saved in every graph from \cG). In this note, we prove that for planar graphs this minimum value is at most 4, and that it is precisely 2 for triangle-free planar graphs.Comment: 15 pages, one reference adde

Multiplicity in the documentation of performance-based artworks: Displaying multi-media documentation in Rebecca Horn’s Body Sculptures at Tate

This is the author accepted manuscript. The final version is available from Taylor & Francis via the DOI in this recordThis paper addresses some of the key issues around authenticity within the ephemeral performance and durable document dichotomy. Engaging with these two artistic practices within the frame of the museum and in the context of displays and exhibitions, this paper considers some of the ways in which the role of performance documentation has been reassessed over the past 20 years. It will focus on the access the document provides to a now-absent ‘performance moment’, the benefit of acquiring and displaying multiple types of documentation, and the experience of the museum visitor within this.Arts and Humanities Research Council (AHRC

Space and value in the contemporary art museum: The journey of a performance document at Tate

This is the author accepted manuscript. The final version is available from Intellect via the DOI in this recordThe space of the museum, rather than being monolithic and heterogeneous, is complex, fluid and fractured. As an institution, its multiple spaces relate to a variety of activities, motivations and attitudes towards the objects it collects, conserves and displays. By using Michel Foucault’s 1967 notion of the ‘heterotopia’ to read the museum as a space of spaces, and focusing on the complex object of the performance document, this article traces the link between the placement of objects in a specific space, and how this can be read as a perspective on their value. In tracing the journey of the Joseph Beuys performance document Four Blackboards (1972) through various spaces at Tate Gallery (now Tate Britain) and Tate Modern, this article will demonstrate those acts of valuation being undertaken over a 50-year period in the institution, and explore how changing value perspectives result in a changing space, both physically and conceptually, for the performance document.Arts and Humanities Research Council (AHRC