In words, we trade in the edge relation of a graph with the function that assigns to each point its set of children.
The IsPath function takes sequence of blocks and edges as an input and checks if the sequence forms a path in the edge relation.
For instance that the reflexive, transitive closure \(E^* x,y)\) of the edge relation of a graph \(G = \langle V,E \rangle\) as the smallest relation satisfying the condition \(E^* x,y)\) is thus definable as \ \text{Fix}((x = y) \vee E x,y) \vee \exists z(R x,z) \wedge R z,y))\).
Similarly, if \(X\) is a graph problem and \(G\) is one of its instances, then the associated structure will be \(\mathcal{A}_G = \langle \{1,2,\ldots,v\},E \rangle\) where we assume that \(G\) has vertices \(\{1,2,\ldots,v\}\) and edge relation \(E \subseteq \{1,2,\ldots,v\} \times \{1,2,\ldots,v\}\).
For instance that the reflexive, transitive closure \(E^* x,y)\) of the edge relation of a graph \(G = \langle V,E \rangle\) as the smallest relation satisfying the condition \[ E^* x,y) \leftrightarrow [(x = y) \vee E x,y) \vee \exists z(E^* x,z) \wedge E^* z,y)] \] \(E^* x,y)\) is thus definable as \ \text{Fix}((x = y) \vee E x,y) \vee \exists z(R x,z) \wedge R z,y))\).
For example, a finite graph can be regarded as a pair (V, E) consisting of a non-empty vertex set V and a symmetric edge relation E on V. The statement that the graph can be properly colored with three colors can be expressed by a second-order sentence: there exist subsets R, G, B that partition V in such a way that two vertices connected by an edge are never the same color.
