Sphere coverings and identifying codes
نویسندگان
چکیده
In any connected, undirected graph G = (V, E), the distance d(x, y) between two vertices x and y of G is the minimum number of edges in a path linking x to y in G. A sphere in G is a set of the form Sr (x) = {y ∈ V : d(x, y) = r}, where x is a vertex and r is a nonnegative integer called the radius of the sphere. We first address in this paper the following question: What is the minimum number of spheres with fixed radius r ≥ 0 required to cover all the vertices of a finite, connected, undirected graph G? We then turn our attention to the Hamming Hypercube of dimension n, and we show that the minimum number of spheres with any radii required to cover this graph is either n or n + 1, depending on the parity of n. We also relate the two above problems to other questions in combinatorics, in particular to identifying codes.
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عنوان ژورنال:
- Des. Codes Cryptography
دوره 70 شماره
صفحات -
تاریخ انتشار 2014