We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call 'spider moves.' We then create a category by modding out by the 2-cells of our 2-category and use the spider moves to show that for finite graphs, this category is a homotopy category in the sense that it satisfies the universal property for localizing homotopy equivalences. We then show that finite stiff graphs form a skeleton of this homotopy category.

Directed graphs have long been used to gain understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Fruct's Theorem. We investigate four inverse semigroups defined over undirected graphs constructed from the notions of subgraph, vertex induced subgraph, rooted tree induced subgraph, and rooted path induced subgraph. We characterize the structure of the semilattice of idempotents and lattice of ideals of these four inverse semigroups. Finally, we prove a characterization theorem that states that every graph has a unique associated inverse semigroup up to isomorphism.

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Elongator dysfunction is increasingly recognized as a contributor to multiple neurodevelopmental and neurodegenerative disorders including familial dysautonomia, intellectual disability, amyotrophic lateral sclerosis, and autism spectrum disorder. Although numerous cellular processes are perturbed in the context of Elongator loss, converging evidence from multiple studies has resolved Elongator's primary function in the cell to the modification of tRNA wobble uridines and the translational regulation of codon-biased genes. Here we characterize H2a.z, encoding the variant H2a histone H2A.Z, as an indirect Elongator target. We further show that canonical Notch signaling, a pathway directed by H2A.Z, is perturbed as a consequence of Elp1 loss. Finally, we demonstrate that hyperacetylation of H2A.Z and other histones via exposure to the histone deacetylase inhibitor Trichostatin A during neurogenesis corrects the expression of Notch3 and rescues the development of sensory neurons in embryos lacking the Elp1 Elongator subunit.

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