Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

The Path to Otopia: an Australian Perspective

This paper is a response to an invitation from SASA to deliver a keynote address on the topic: "The History of Innovative Organic Knowledge: Past, Present (and Future?)” to the Soil Association of South Australia (SASA) on the occasion of the launching of the SASA Historical Research Archive at the State Library of South Australia, Adelaide. It identifies three waves of organic advocacy in Australia. It describes the author's recently published research on the Australian Organic Farming and Gardening Society (1944-1955), the world's first society to call itself an "organic farming" society, the first society to publish an organic journal (the "Organic Farming Digest"), and the first society to publish a set of organic agriculture principles. Looking to the future, the term "Otopia" is coined to describe a state of 100% organic agriculture. At the historical rate of growth exhibited by the organic sector (data available for the past 8 years), it will take 584 years to reach a global state of Otopia if we assume arithmetic growth (of 27.1% pa), or 27 years if we assume compounding growth (of 16.4% pa)

Revolutionaries and spies: Spy-good and spy-bad graphs

We study a game on a graph $G$ played by $r$ {\it revolutionaries} and $s$ {\it spies}. Initially, revolutionaries and then spies occupy vertices. In each subsequent round, each revolutionary may move to a neighboring vertex or not move, and then each spy has the same option. The revolutionaries win if $m$ of them meet at some vertex having no spy (at the end of a round); the spies win if they can avoid this forever. Let $\sigma(G,m,r)$ denote the minimum number of spies needed to win. To avoid degenerate cases, assume |V(G)|\ge r-m+1\ge\floor{r/m}\ge 1. The easy bounds are then \floor{r/m}\le \sigma(G,m,r)\le r-m+1. We prove that the lower bound is sharp when $G$ has a rooted spanning tree $T$ such that every edge of $G$ not in $T$ joins two vertices having the same parent in $T$. As a consequence, \sigma(G,m,r)\le\gamma(G)\floor{r/m}, where $\gamma(G)$ is the domination number; this bound is nearly sharp when $\gamma(G)\le m$. For the random graph with constant edge-probability $p$, we obtain constants $c$ and $c'$ (depending on $m$ and $p$) such that $\sigma(G,m,r)$ is near the trivial upper bound when $r<c\ln n$ and at most $c'$ times the trivial lower bound when $r>c'\ln n$. For the hypercube $Q_d$ with $d\ge r$, we have $\sigma(G,m,r)=r-m+1$ when $m=2$, and for $m\ge 3$ at least $r-39m$ spies are needed. For complete $k$-partite graphs with partite sets of size at least $2r$, the leading term in $\sigma(G,m,r)$ is approximately $\frac{k}{k-1}\frac{r}{m}$ when $k\ge m$. For $k=2$, we have \sigma(G,2,r)=\bigl\lceil{\frac{\floor{7r/2}-3}5}\bigr\rceil and \sigma(G,3,r)=\floor{r/2}, and in general $\frac{3r}{2m}-3\le \sigma(G,m,r)\le\frac{(1+1/\sqrt3)r}{m}$.Comment: 34 pages, 2 figures. The most important changes in this revision are improvements of the results on hypercubes and random graphs. The proof of the previous hypercube result has been deleted, but the statement remains because it is stronger for m<52. In the random graph section we added a spy-strategy resul

How to Guard a Graph?

We initiate the study of the algorithmic foundations of games in which a set of cops has to guard a region in a graph (or digraph) against a robber. The robber and the cops are placed on vertices of the graph; they take turns in moving to adjacent vertices (or staying). The goal of the robber is to enter the guarded region at a vertex with no cop on it. The problem is to find the minimum number of cops needed to prevent the robber from entering the guarded region. The problem is highly non-trivial even if the robber's or the cops' regions are restricted to very simple graphs. The computational complexity of the problem depends heavily on the chosen restriction. In particular, if the robber's region is only a path, then the problem can be solved in polynomial time. When the robber moves in a tree (or even in a star), then the decision version of the problem is NP-complete. Furthermore, if the robber is moving in a directed acyclic graph, the problem becomes PSPACE-complet

The Surprising History and Geography of the First "Organic Farming" Association

Readers of narratives of the history of organic farming in Australia will be familiar with such accounts beginning in the "1980s". In questing after the earliest organic farming society, and more particularly in pursuing the spread of the "organic" meme from its 1940 birth in Britain, it was therefore a great surprise to uncover the Australian Organic Faming and Gardening Society (AOFGS) founded in October 1944. This appears to be the world's first "organic farming" association. It also resets the organic clock for Australia back by four decades. Here was an association, pre-dating the UK Soil Association by two years, formed half a world away from the birthplace of "organic", in a country at war, under food rationing, and with its workforce under Manpower regulations. Yet organic farming principles were clearly articulated by this Society, perhaps as clearly articulated as they have ever been, and particularised for Australia. The Society was constrained from publishing their journal due to wartime constraints on paper. The first appearance of the Organic Farming Digest (OFD) was in April 1946. Thereafter, for nearly a decade, the Society regularly published a journal, with the last issue appearing in 1954. This paper explores the Society; its principles; its journals; its people; its interactions with key organic figures of the time including Ehrenfried Pfeiffer, Eve Balfour, Albert Howard, and Jerome Rodale; its Australian contributors from five states, including Colonel Harold White and Professor Sir Stanton Hicks; its progress and ultimately its demise; and touch on how this history became lost