Characterization of co-blockers for simple perfect matchings in a convex geometric graph

Consider the complete convex geometric graph on $2m$ vertices, $CGG(2m)$, i.e., the set of all boundary edges and diagonals of a planar convex $2m$-gon $P$. In [C. Keller and M. Perles, On the Smallest Sets Blocking Simple Perfect Matchings in a Convex Geometric Graph], the smallest sets of edges that meet all the simple perfect matchings (SPMs) in $CGG(2m)$ (called "blockers") are characterized, and it is shown that all these sets are caterpillar graphs with a special structure, and that their total number is $m \cdot 2^{m-1}$. In this paper we characterize the co-blockers for SPMs in $CGG(2m)$, that is, the smallest sets of edges that meet all the blockers. We show that the co-blockers are exactly those perfect matchings $M$ in $CGG(2m)$ where all edges are of odd order, and two edges of $M$ that emanate from two adjacent vertices of $P$ never cross. In particular, while the number of SPMs and the number of blockers grow exponentially with $m$, the number of co-blockers grows super-exponentially.Comment: 8 pages, 4 figure

On Convex Geometric Graphs with no k+1k+1 Pairwise Disjoint Edges

A well-known result of Kupitz from 1982 asserts that the maximal number of edges in a convex geometric graph (CGG) on $n$ vertices that does not contain $k+1$ pairwise disjoint edges is $kn$ (provided $n>2k$). For $k=1$ and $k=n/2-1$, the extremal examples are completely characterized. For all other values of $k$, the structure of the extremal examples is far from known: their total number is unknown, and only a few classes of examples were presented, that are almost symmetric, consisting roughly of the $kn$ "longest possible" edges of $CK(n)$, the complete CGG of order $n$. In order to understand further the structure of the extremal examples, we present a class of extremal examples that lie at the other end of the spectrum. Namely, we break the symmetry by requiring that, in addition, the graph admit an independent set that consists of $q$ consecutive vertices on the boundary of the convex hull. We show that such graphs exist as long as $q \leq n-2k$ and that this value of $q$ is optimal. We generalize our discussion to the following question: what is the maximal possible number $f(n,k,q)$ of edges in a CGG on $n$ vertices that does not contain $k+1$ pairwise disjoint edges, and, in addition, admits an independent set that consists of $q$ consecutive vertices on the boundary of the convex hull? We provide a complete answer to this question, determining $f(n,k,q)$ for all relevant values of $n,k$ and $q$.Comment: 17 pages, 9 figure

The Production of Just Space: Climate Change and the Future of the New York City Housing Authority

The scientific community is all but certain that the planet is warming as a direct result of human interference with natural climatic systems. Climate change will affect everyone and everything on the planet but these effects will not be distributed equally or in the same way. It is clear that the most marginalized populations on the planet - the (urban) poor, women, children, people of color, the LGBTQ community and the elderly - will be most severely affected. Historically, national and international progress has been inadequate in addressing the profound issues surrounding climate change. Substantive progress is being made, however, in cities, especially in New York City. This thesis seeks to understand the idea that progress at a local, community level can affect city-wide policies and, potentially, larger scale (inter)national climate goals. The New York City Housing Authority’s (NYCHA) Resident Green Committees (RGCs) are used as a lens to examine how pressure from communities can affect change in a climate agenda. Through a review of critical urban and spatial theory; interviews with RGC leaders, NYCHA officials, and public housing residents; and an analysis of the agendas and projects taking place at different housing projects and in NYC housing activism circles, an understanding of the ways in which socio-spatial justice influences our collective future will be completed

Assessing Logging Concession Road Prevalence as a Mechanism for Land Use Change in the Republic of the Congo

Honors (Bachelor's)EnvironmentUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/113081/1/Honors Thesis Perles Small.pdfDescription of Honors Thesis Perles Small.pdf : Thesi