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A graph $G$ is said to be a model of a first-order sentence $\varphi$ if $G$ satisfies $\varphi$. Now let $\varphi(x_1,...,x_r)$ be a first order formula with free variables $x_1,...,x_r$. What is the standard terminology for a tuple of vertices $(v_1,...,v_r)$ of $G$ such that $G$ satisfies the formula $\varphi(v_1,...,v_r)$ obtained by substituting $x_i$ with $v_i$?

What I'm having trouble is to find the right terminology to connect $G$ to the assignment. The only thing I can think now is the following.

"$(v_1,...,v_r)$ is a satisfying assignment for $\varphi$ in $G$."

But maybe there is a shorter way.

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I think that's often called a "variable assignment", since it assigns to each variable a value (a vertex in the graph, in your case).

If the graph is equipped with a set of such tuples, these might be considered to be hyperedges, i.e. edges connected to an arbitrary number of vertices (not necessarily two of them).

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  • $\begingroup$ Thanks. I'm fine with the fact that it is an assignment. What I'm having trouble is to find the right terminology to connect $G$ to the assignment. For instance, would the standard be to say that $(v_1,...,v_r)$ is a satisfying assignment for $\varphi$ in $G$? $\endgroup$
    – verifying
    Mar 1 '19 at 20:28
  • $\begingroup$ @verifying I would understand that. You could even say that $(v_1,\ldots,v_r)$ satisfies $\varphi$ for short. I can't guarantee that's the most standard way, but I find that very reasonable and clear. $\endgroup$
    – chi
    Mar 1 '19 at 23:00

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