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- W1976360097 abstract "A family F of functions from ℝn to R is k-point separating if, for every k-subset S of ℝn, there is a function f∈F such that f is one-to-one on S. The paper shows that, if the functions are required to be linear (or smooth), then a minimum k-point separating family F has cardinality n(k−1). In the linear case, this result is extended to a larger class of fields including all infinite fields as well as some finite fields (depending on k and n). Also, some partial results are obtained for continuous functions on ℝn, including the case when k is infinite. The proof of the main result is based on graph theoretic results that have some interest in their own right. Say that a graph is an n-tree if it is a union of n edge-disjoint spanning trees. It is shown that every graph with k⩾2 vertices and n(k−1) edges has a non-trivial subgraph which is an n-tree. A determinantal criterion is also established for a graph with k vertices and n(k−1) edges to be an n-tree." @default.
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- W1976360097 date "1998-10-01" @default.
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- W1976360097 title "An Application of Spanning Trees to k -Point Separating Families of Functions" @default.
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- W1976360097 doi "https://doi.org/10.1112/s0024610798006437" @default.
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