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29 March 2024
 
  » arxiv » 1106.3632

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Minimal resolving sets for the hypercube
Ashwin Ganesan ;
Date 18 Jun 2011
AbstractLet $G=(V,E)$ be a simple, undirected graph. An ordered subset $S = {s_1,s_2,...,s_k} subseteq V$ of vertices is a resolving set for $G$ if the vertices of $G$ are distinguishable by their vector of distances to the vertices in $S$. The $n$-dimensional hypercube $Q^n$ is the graph whose vertices are the $2^n$ 0-1 sequences of length $n$, with two vertices being adjacent whenever the corresponding two sequences differ in exactly one coordinate. In cite{Erdos:Renyi:1963} it was shown that the vertices ${11..., 011..,$ $1011...1,..., 111...101}$ form a resolving set for the hypercube. The purpose of this note is to prove that a proper subset of that set, namely the vertices ${011...1, 1011...1,$ $..., 111...101}$, also forms a resolving set for the hypercube for all $n ge 5$.
Source arXiv, 1106.3632
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