Abstract. The zero forcing number, maximum nullity and path cover number of a (simple, undirected) graph are parameters that are important in the study of minimum rank problems. We investigate the effects on these graph parameters when an edge is subdivided to obtain a so-called edge subdivision graph. An open question raised by Barrett et al. is answered in the negative, and we provide additional evidence for an affirmative answer to another open question in that paper [W. Barrett, R. Bowcutt, M. Cutler, S. Gibelyou, and K. Owens. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530-563, 2009.]. It is shown that there is an independent relationship between the change in maximum nullity and zero forcing number caused by subdividing an edge once. Bounds on the effect of a single edge subdivision on the path cover number are presented, conditions under which the path cover number is preserved are given, and it is shown that the path cover number and the zero forcing number of a complete subdivision graph need not be equal.

We approach manifold topology by examining configurations of finite subsets of manifolds within the homotopy-theoretic context of ∞-categories by way of stratified spaces. Through these higher categorical means, we identify the homotopy types of such configuration spaces in the case of Euclidean space in terms of the category Θn.Contents 2. The exit-path ∞-category of the unital Ran Space 2.1. Proving Exit Ran u (M ) is an ∞-category 2.2. The main result, Theorem 2.2.1 3. Part 1 of the proof of Theorem 2.2.1: Refining Exit Ran u (R n ) 3.1. The exit-path ∞-category of the refined unital Ran space of R n 3.2. Proving the exit-path ∞-category of the refined unital Ran space of R n is an ∞-category 3.3. Identifying the exit-path ∞-category of the refined unital Ran space of R n as Θ act n 4. Part 2 of the proof of Theorem 2.2.1: Localizing to Exit Ran u (R n ) 4.1. Identifying the space of objects of the localization of Lemma 4.0.2 4.2. Identifying the space of morphisms of the localization of Lemma 4.0.2 4.3. Proving Lemma 4.0.8 5. A corollary to Theorem 2.2.1: Identifying Exit Ran(R n ) combinatorially 5.1. The exit-path ∞-category of the Ran space of R n 5