A family of sets ℋ is ideal if the polyhedron {x ≥ 0 : Σi∈Sxi ≥ 1, ∀S ∈ ℋ} is integral. Consider a graph G with vertices s, t. An odd st-walk is either an odd st-path or the union of an even st-path and an odd circuit that share, at most, one vertex. Let T be a subset of vertices of even cardinality. An st-T-cut is a cut of the form δ(U) where |U ∩ T| is odd and U contains exactly one of s or t. We give excluded minor characterizations for when the families of odd st-walks and st-T-cuts (represented as sets of edges) are ideal. As a corollary, we characterize which extensions and coextensions of graphic and cographic matroids are 1-flowing.
This opening chapter details how this book emerged and developed, its key themes and structure. In so doing, the chapter will discuss intersectionality, multi-disciplinarity and why this is a timely and important edited collection. The chapter discusses how it is important that the intersections of ageing, gender and sexualities are considered together, alongside other sources of social division and identity.
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