Denote a directed graph of the node-set $V$ and the directed-edge set $E$ by $G=(V,E)$. Denote a summarized directed graph of the compressed node-set $C$ and the directed relation set $R$. In this work, both $G$ and $H$ are simple. We distinguish terms in both graphs shown in Table 1. A node-compression is a function $\varphi V:V\u2192C$ that assigns a vertex $xi\u2208V$ to a compressed node $cI\u2208C$. In this work, $\varphi V$ is surjective. Note that we do not require all vertices to belong to an associated compressed node as opposed to the graph partition problem, that is, the union of members in an identified compressed node does not need to be the full set of vertices. An edge-compression is a function $\varphi E:E\u2192R$. We say an edge-compression, $\varphi E$, is induced from a node-compression $\varphi V$ if $\varphi V(xi)=\varphi V(xi')$, $\varphi V(xj)=\varphi V(xj')$, implies $\varphi E(eij)=\varphi E(ei'j')$, $\u2200i,j,i',j'$, that is, vertices assigned to the same compressed node admit the same compressed relation. Hence, we can write $\varphi E(eij)=rIJ$, $\u2200\varphi V(xi)=cI,\varphi V(xj)=cJ$. In this work, we only consider the edge-compression induced from the node-compression.

Table 1:

original graph: $G$ | vertices: $xi\u2208V$ | edges: $eij\u2208E$ |

summarized graph: $H$ | compressed nodes: $cI\u2208C$ | compressed relations: $rIJ\u2208R$ |

original graph: $G$ | vertices: $xi\u2208V$ | edges: $eij\u2208E$ |

summarized graph: $H$ | compressed nodes: $cI\u2208C$ | compressed relations: $rIJ\u2208R$ |

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