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I recently studied about this construction of a spanning graph for a set of points in $\mathbb{R}^2$ and I feel like I'm not really understanding why this construction should work.

First there are some required definitions:

Two sets $A,B$ are said to be well seperated w.r.t to $s>0$, if there exists some $r\in \mathbb R$ such that both $A$ and $B$ are contained in separate balls of radius $r$ $B_A , B_B$ and $d(B_A, B_B) \geq sr$.

WSPD:

Let $S\subset \mathbb R^2$.

$\{\{A_1,B_1\},...,\{A_m,B_m\}\}$ is a WSPD for $S$ w.r.t $s>0$ if $A_i,B_i$ are well seperated w.r.t $s$ and for all $p,q\in S$ there exists a unique $i$ such that $p\in A_i, q\in B_i$ or $q\in A_i, p\in B_i$.

In case something is not clear with the way I've defined things, it is all stated here

The algorithm that uses WSPD to construct a t-spanner, uses $t=\frac{s+4}{s-4}$,

Let $S\subset \mathbb R^2$ and a WSPD for $S: \{\{A_1,B_1\},...,\{A_m,B_m\}\}$ let $G$ be a graph on $S$ with $E=\{\{a_i,b_i\}: a_i\in A_i, b_i\in B_i, 1\leq i\leq m\}$.

Basically what it does is consider a WSPD, and for every pair it picks an edge for the spanner arbitrarily. What I don't understand is why is it not possible for the output graph to have even isolated vertices? I mean, couldn't it be that some point $q\in S$ will never be picked for an edge?

Any insights on this algorithm would be appreciated, especially on what I asked with regards to it not having isolated vertices.

Thanks in advance.

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  • $\begingroup$ Why do you say that the spanner picks one edge for every pair? Given a pair $\{A_i, B_i\}$, all pairs $\{a_i, b_i\}$ with $a_i \in A_i$ and $b_i \in B_i$ satisfy the condition in the definition of $E$, therefore they should all be included. $\endgroup$
    – Steven
    Aug 26, 2022 at 16:37

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