The problem Cover(H) asks whether an input graph G covers a fixed graph H (i.e., whether there exists a homomorphism G → H which locally preserves the structure of the graphs). Complexity of this problem has been intensively studied. In this paper, we consider the problem PlanarCover(H) which restricts the input graph G to be planar. PlanarCover(H) is polynomially solvable if Cover(H) belongs to P, and it is even trivially solvable if H has no planar cover. Thus the interesting cases are when H admits a planar cover, but Cover(H) is NP-complete. This also relates the problem to the long-standing Negami Conjecture which aims to describe all graphs having a planar cover. Kratochvíl asked whether there are non-trivial graphs for which Cover(H) is NP-complete but PlanarCover(H) belongs to P. We examine the first nontrivial cases of graphs H for which Cover(H) is NP-complete and which admit a planar cover. We prove NP-completeness of PlanarCover(H) in these cases.
Red eyes are a common presentation to ophthalmic services. This article aims to increase understanding of some of the more common, non traumatic ‘red eyes’. It describes the presenting features, disease process and evidence-based care and treatment of this group of patients and includes some ‘time out’ activities. Undertaking the reflective work suggested by these activities will assist the clinician in linking the theory presented here to patients seen in the practice environment and thus enable them to give effective, evidence-based care to this client group.
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