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Efficient open dominating sets in bipartite Cayley graphs are characterized in terms of covering projections. Necessary and sufficient conditions for the existence of efficient open dominating sets in certain circulant Harary graphs are given. Chains of efficient dominating sets, and of efficient open dominating sets, in families of circulant graphs are described as an application.

Path planning can be subject to different types of optimization. Some years ago a German researcher, U. Leuthäusser, proposed a new variational method for reducing most types of optimization criteria to one and the same: minimization of path length. This can be done by altering the Riemannian metric of the domain, so that optimal paths (with respect to whatever criterion) are simply seen as shortest. This method offers an extra feature, which has not been exploited so far: it admits direction–dependent criteria. In this paper we make this feature explicit, and apply it to two different anisotropic settings. One is that of different costs for different directions: E.g. the situation of a countryside scene with ploughed fields. The second is dependence on oriented directions, which is called here "strong" anisotropy: the typical scene is that of a hill side. A covering projection solves the additional difficulty. We also provide some experimental results on synthetic data.

It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the \emph{fundamental progroupoid}, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and can not be replaced by a localic groupoid. The classifying topos in not any more a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver July 2004., 19 pages

In this paper, for a given space X , a structural category C , and a faithful functor η from C to the category of spaces, we introduce a notion of ( C , η)-bundle which contains as particular cases, the notions of covering space, of overlaying space (introduced by Fox), of suspension foliation and other well-known topological structures. The new notion allows us to use sheaf theory and category theory in order to obtain some classification theorems which appear in terms of equivalences of categories. We prove that the category ( C , η)-bundle(X) of ( C , η)-bundles over X is equivalent to the category pro(πCX, C ) , which is determined by the fundamental groupoid of X and the structural category C . As particular cases we obtain the standard classification of covering spaces, Fox's classification theorem for overlays with a finite number of leaves and the standard classification of suspension foliations. This paper illustrates the importance of the fundamental progroupoid of a space X , which plays in shape theory the role of the standard fundamental groupoid. If the space X satisfies some additional properties, the progroupoid πCX can be reduced to a surjective progroup, a groupoid or a group. In some cases a surjective progroupoid can be replaced by a topological prodiscrete group. In all these cases the category pro(πCX, C ) also reduces to well-known categories.