We prove upper and lower bounds on the size of the largest square grid graph that is a subgraph, minor, or shallow minor of a graph in the form of a larger square grid from which a specified number of vertices have been deleted. Our bounds are tight to within constant factors. We also provide less-tight bounds on analogous problems for higher-dimensional grids.

A minor-closed class of graphs is a set of labelled graphs which is closed under isomorphism and under taking minors. For a minor-closed class G, let gn be the number of graphs in G which have n vertices. The growth constant of G is γ = lim sup(gn/n!) . We study the properties of the set Γ of growth constants of minor-closed classes of graphs. Among other results, we show that Γ does not contai...

Papillary cystadenoma is a rare, benign salivary gland tumor which is well-circumscribed, containing cystic cavities with intraluminal papillary projections. Only 19 cases arising within minor salivary glands (MnSG) from the oral cavity sites have been reported in the English literature (PubMed 1958-2014). We report 11 new cases of MnSG papillary cystadenomas in conjunction with a review of the...

This paper focuses on a number of American stories which helped educate people and bring about legal or social change. There are many stories which caused major or minor legal and political change, particularly, in the United States. Some of them are written by Herman Melville, Harriet Beecher Stowe, Upton Sinclair and Sidney Kingsley. After the publication of White Jacket by Melville the novel...

amelogenesis imperfecta is a group of genetic disorders that affects both the morphology and quality of tooth structure. although the disease entity is primarily associated with abnormalities of dental and oral structures, it has been reported to be associated with a few syndromes. a 9-year-old girl with minor thalassemia referred to the department of pediatric dentistry of the mashhad faculty ...