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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 ϕV:VC that assigns a vertex xiV to a compressed node cIC. In this work, ϕV 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 ϕE:ER. We say an edge-compression, ϕE, is induced from a node-compression ϕV if ϕV(xi)=ϕV(xi'), ϕV(xj)=ϕV(xj'), implies ϕE(eij)=ϕE(ei'j'), i,j,i',j', that is, vertices assigned to the same compressed node admit the same compressed relation. Hence, we can write ϕE(eij)=rIJ, ϕV(xi)=cI,ϕV(xj)=cJ. In this work, we only consider the edge-compression induced from the node-compression.

Table 1:

Term Comparison between Original Graph and Summarized Graph.

original graph: G vertices: xiV edges: eijE 
summarized graph: H compressed nodes: cIC compressed relations: rIJR 
original graph: G vertices: xiV edges: eijE 
summarized graph: H compressed nodes: cIC compressed relations: rIJR 

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