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Let $(X,d)$ be a metric space, and suppose $S\subseteq X$ is a finite subset in which all pairwise distances are distinct (formal definition here).

If $x\in S$ and $k$ is a non-negative integer with $k<|S|$, we define $S_k(x)\subseteq S$ to be the set of the $k$ nearest members of $S$ to $x$, other than $x$ itself. We define a graph $G(k,S)$ by setting $V(G(k,S)) = S$, and $$E(G(k,S)) = \big\{\{x,y\}: x\neq y \in S \text{ and } y\in S_k(x) \text{ and }x\in S_k(y)\big\}.$$ So the points $x,y$ are connected by an edg if and only if $y$ belongs to the $k$ nearest points of $x$, and vice versa.

We say that a metric space $(X,d)$ is graph-universal if for every finite graph $G$ there is $S\subseteq X$ with pairwise distinct distances, and a non-negative integer $k<|S|$ such that $G\cong G(k,S)$. For example, $\mathbb{R}$ with the Euclidean distance is not graph-universal as we cannot find a set $S$ of $4$ real points and $k<4$ such that $C_4 \cong G(k, S)$.

Question. If $\{0,1\}^{<\omega}$ denotes the set of functions $f:\omega\to\{0,1\}$ that are eventually $0$, and endow it with the Hamming distance $d_H$, is $(\{0,1\}^{<\omega}, d_H)$ graph-universal?

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    $\begingroup$ How would you represent a star in any metric space? The center has $n-1$ neighbours, so we'd have $k = n-1$. But, unless I'm missing something, $G(n-1,S)$ for $|S| = n$ always gives a complete graph. $\endgroup$ – Florian Lehner Oct 17 '18 at 13:56
  • $\begingroup$ @FlorianLehner that's correct, thanks for this observation! I will delete the question - unless you want to post your argument as an answer -- you are very welcome to! I'll accept and upvote it. $\endgroup$ – Dominic van der Zypen Oct 18 '18 at 6:50
  • $\begingroup$ I just posted my comment as an answer. $\endgroup$ – Florian Lehner Oct 18 '18 at 15:23
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It is not possible to embed any non-complete graph with a universal vertex into any metric space in this way.

A universal vertex in an $n$-vertex graph has degree $n-1$ and thus $k = n-1$ in any representation of such a graph. But if $|S| = n$ and $k = n-1$, then $G(k,S)$ is a complete graph on $n$ vertices.

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