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I'm reading the article "What can be computed locally?" by Naor & Stockmeyer and I struggle to understand the definition of an LCL they gave. Here is an extract: (page 2)

An Locally Checkable Labelling (LCL) $\mathcal{L}$ consist of a positive integer $r$, a set $\Sigma$ of input labels, a set $\Gamma$ of output labels and a set $\mathcal{C}$ of of locally consistent labels. Given a graph $G$ and $\lambda: V\mapsto \Sigma \times \Gamma$, we say that $\lambda$ is $\mathcal{L}$-legal if $\forall u \in V$, there exists $(H,s) \in \mathcal{C}$ and an isomorphism $\pi$ mapping $B(u,r)$ to $H$ such that $\pi(u) = s$ and $\pi$ respect the labelling, i.e. for all label-pair $w$, equals the label pair $\pi(w)$.

Note: $(H,s)$ is a graph centered in $s$, a.k.a. $s \in H$.

I don't really understand some part of this definition, but mostly what $\mathcal{C}$ looks like. I tried to think of a simple exemple: classic vertex coloration where you don't allow to neighbors to have the same colors.

Suppose we need $c$ colors; so $\Gamma = [c]$. Clearly $r=1$ as it is sufficient for nodes to check up to their direct neighbors. Now I'm not sure what $\Sigma$ can look like; I suspect in the case of coloration it is not really needed as this problem is too simple? As for $\mathcal{C}$ my guess is that it should include the informations relative to all the vertices and their direct neighbors, but I don't see how and what $\pi$ fits in all this.

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  • $\begingroup$ "I don't really understand ..." is hardly a research level question, which is what cstheory targets. $\endgroup$
    – Kai
    Aug 30, 2023 at 15:59
  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Kai
    Aug 30, 2023 at 16:00

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