Making deeper sense in the midst of great busyness: a study of and with third sector CEOs

This paper provides the background to a study from which sample findings will be presented and discussed at the ARVOVA 2008 conference in Philadelphia. It explains the origins and aims of a continuing collaborative inquiry with third sector Chief Executives, gives a summary history of the project, and describes the methodological issues involved in doing 'twin-track' research – that is, research aimed at producing both practice knowledge and formal theory

On the multipacking number of grid graphs

In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical domination where selected vertices may have different domination powers. The minimum cost of a dominating broadcast in a graph $G$ is denoted $\gamma_b(G)$. The dual of this problem is called multipacking: a multipacking is a set $M$ of vertices such that for any vertex $v$ and any positive integer $r$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $M$ . The maximum size of a multipacking in a graph $G$ is denoted mp(G). Naturally mp(G) $\leq \gamma_b(G)$. Earlier results by Farber and by Lubiw show that broadcast and multipacking numbers are equal for strongly chordal graphs. In this paper, we show that all large grids (height at least 4 and width at least 7), which are far from being chordal, have their broadcast and multipacking numbers equal

'Unhappily in love with God': conceptions of the divine in the poetry of Geoffrey Hill, Les Murray and R.S Thomas

This thesis looks at the poetry of three markedly different contemporary poets, Geoffrey Hill, Les Murray and R. S. Thomas. They are linked by at least tacit belief in Christianity and the Christian world-view, and this belief shapes everything they write, whether explicitly 'religious' or otherwise. My focus throughout the thesis is on Hill, Murray and Thomas's differing conceptions of God, and my explorations of their poetic and religious stances take God as both their starting point and destination. The opening chapter is a general introduction to the possibilities of religious poetry in the modern world, before turning, in chapter two, to Hill, Murray and Thomas themselves and an identification of their religious concerns and sensibilities. The remaining thematic chapters concern themselves with Hill and Murray's explorations of suffering and evil, post-1945; the place of humour and laughter in the religious visions of Murray and Hill; Murray's remarkable sequence of animal poems, 'Presence'; and the figure of Christ in the poetry of Thomas. I conclude with a discussion of T. S. Eliot's misgivings concerning religious poetry, and how Hill, Murray and Thomas avoid writing the limited poetry he identifies. My method throughout is to base my discussion of these three poets on close readings of their individual poems

List homomorphism problems for signed graphs

We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph $(G,\sigma)$, equipped with lists $L(v) \subseteq V(H), v \in V(G)$, of allowed images, to a fixed target signed graph $(H,\pi)$. The complexity of the similar homomorphism problem without lists (corresponding to all lists being $L(v)=V(H)$) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. We illustrate this difficulty by classifying the complexity of the problem when $H$ is a tree (with possible loops). The tools we develop will be useful for classifications of other classes of signed graphs, and we illustrate this by classifying the complexity of irreflexive signed graphs in which the unicoloured edges form some simple structures, namely paths or cycles. The structure of the signed graphs in the polynomial cases is interesting, suggesting they may constitute a nice class of signed graphs analogous to the so-called bi-arc graphs (which characterize the polynomial cases of list homomorphisms to unsigned graphs).Comment: various changes + rewritten section on path- and cycle-separable graphs based on a new conference submission (split possible in future

A Dichotomy Theorem for Circular Colouring Reconfiguration

The "reconfiguration problem" for circular colourings asks, given two $(p,q)$-colourings $f$ and $g$ of a graph $G$, is it possible to transform $f$ into $g$ by changing the colour of one vertex at a time such that every intermediate mapping is a $(p,q)$-colouring? We show that this problem can be solved in polynomial time for $2\leq p/q <4$ and is PSPACE-complete for $p/q\geq 4$. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings.Comment: 22 pages, 5 figure

Building blocks for the variety of absolute retracts

AbstractGiven a graph H with a labelled subgraph G, a retraction of H to G is a homomorphism r:H→G such that r(x)=x for all vertices x in G. We call G a retract of H. While deciding the existence of a retraction to a fixed graph G is NP-complete in general, necessary and sufficient conditions have been provided for certain classes of graphs in terms of holes, see for example Hell and Rival.For any integer k⩾2 we describe a collection of graphs that generate the variety ARk of graphs G with the property that G is a retract of H whenever G is a subgraph of H and no hole in G of size at most k is filled by a vertex of H. We also prove that ARk⊂NUFk+1, where NUFk+1 is the variety of graphs that admit a near unanimity function of arity k+1

Characterizing Circular Colouring Mixing for pq<4\frac{p}{q}<4

Given a graph $G$, the $k$-mixing problem asks: Can one obtain all $k$-colourings of $G$, starting from one $k$-colouring $f$, by changing the colour of only one vertex at a time, while at each step maintaining a $k$-colouring? More generally, for a graph $H$, the $H$-mixing problem asks: Can one obtain all homomorphisms $G \to H$, starting from one homomorphism $f$, by changing the image of only one vertex at a time, while at each step maintaining a homomorphism $G \to H$? This paper focuses on a generalization of $k$-colourings, namely $(p,q)$-circular colourings. We show that when $2 < \frac{p}{q} < 4$, a graph $G$ is $(p,q)$-mixing if and only if for any $(p,q)$-colouring $f$ of $G$, and any cycle $C$ of $G$, the wind of the cycle under the colouring equals a particular value (which intuitively corresponds to having no wind). As a consequence we show that $(p,q)$-mixing is closed under a restricted homomorphism called a fold. Using this, we deduce that $(2k+1,k)$-mixing is co-NP-complete for all $k \in \mathbb{N}$, and by similar ideas we show that if the circular chromatic number of a connected graph $G$ is $\frac{2k+1}{k}$, then $G$ folds to $C_{2k+1}$. We use the characterization to settle a conjecture of Brewster and Noel, specifically that the circular mixing number of bipartite graphs is $2$. Lastly, we give a polynomial time algorithm for $(p,q)$-mixing in planar graphs when $3 \leq \frac{p}{q} <4$.Comment: 21 page

SMILE: the creation of space for interaction through blended digital technology

Interactive Learning Environments at Sussex University is a course in which students are given mobile devices (XDAs) with PDA functionality and full Internet access for the duration of the term. They are challenged to design and evaluate learning experiences, both running and evaluating learning sessions that involve a blend of technologies. Data on technology usage was collected via backups, email and web-site logging as well as video and still photography of student-led sessions. Initial analysis indicates that large amounts of technical support, solid pedagogical underpinning and a flexible approach to both delivery context and medium are essential. The project operated under the acronym SMILE – Sussex Mobile Interactive Learning Environment