Branch-depth: Generalizing tree-depth of graphs

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 34 pages, 2 figure

Branch-depth: Generalizing tree-depth of graphs

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 36 pages, 2 figures. Final versio

Finding Australia’s social enterprise sector: final report

Executive Summary Social enterprises are organisations that: Are led by an economic, social, cultural, or environmental mission consistent with a public or community benefit; Trade to fulfil their mission; Derive a substantial portion of their income from trade; and Reinvest the majority of their profit/surplus in the fulfilment of their mission. This document reports on the research findings of the Finding Australia’s Social Enterprise Sector (FASES) project. FASES is a joint initiative of Social Traders and the Australian Centre for Philanthropy and Nonprofit Studies, Queensland University of Technology. It is a first attempt to identify the range and scope of social enterprises in Australia. The methodology utilised in this research included: a review of existing literature and methods of social enterprise mapping; establishment of a project website and preliminary discussion paper to stimulate public engagement with defining and identifying Australian social enterprises, which resulted in four online responses to the discussion paper and 157 nominations of social enterprises to be included in the research; a series of workshops and interviews with 34 key informants to assist in defining social enterprise for the purposes of the research; identification of the social enterprise population through a combination of web and media review, review of existing databases and feedback through the project website; development and administration of an online survey; and collation and analysis of secondary data. Five hundred and thirty-nine organisations commenced the online survey, of which 365 were valid social enterprises according to our definition. Based on pre-existing research data and information from our survey, we estimate that there are up to 20 000 Australian social enterprises. This estimate takes into account that some not for profit organisations have multiple business ventures, and that not all social enterprises are incorporated as not for profits. Our survey results suggest that the Australian social enterprise sector is mature, sustainable and internally diverse with regard to mission and organisational structure. Amongst the 365 survey respondents, 73% had been operational for at least five years, and 62% were at least 10 years old. Australian social enterprises seek to fulfil a diversity of missions and serve a wide variety of beneficiaries. As a whole, the dominant foci of our survey respondents were on creating opportunities for people to participate in their community, and on finding new solutions to social, environmental, cultural and economic problems. Australian social enterprises operate in every industry of our economy. Our survey data suggest that they trade predominantly in local and regional markets and focus on fulfilling their missions at local and regional goals. However, some social enterprises operate in international markets and seek to respond to missions of international scope

Matt O\u27Keefe to Lead Center for Manufacturing Excellence

Expanded undergraduate program, new graduate program among goals for new executive directo