Quotient graph
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In graph theory, a quotient graph Q of a graph G is a graph whose vertices are blocks of a partition of the vertices of G and where block B is adjacent to block C if some vertex in B is adjacent to some vertex in C with respect to the edge set of G.[1] In other words, if G has edge set E and vertex set V and R is the equivalence relation induced by the partition, then the quotient graph has vertex set V/R and edge set {([u]R, [v]R) | (u, v) ∈ E(G)}.
More formally, a quotient graph is a quotient object in the category of graphs. The category of graphs is concretizable – mapping a graph to its set of vertices makes it a concrete category – so its objects can be regarded as "sets with additional structure", and a quotient graph corresponds to the graph induced on the quotient set V/R of its vertex set V. Further, there is a graph homomorphism (a quotient map) from a graph to a quotient graph, sending each vertex or edge to the equivalence class that it belongs to. Intuitively, this corresponds to "gluing together" (formally, "identifying") vertices and edges of the graph.
A graph is trivially a quotient graph of itself (each block of the partition is a single vertex), and the graph consisting of a single point is the quotient graph of any non-empty graph (the partition consisting of a single block of all vertices). The simplest non-trivial quotient graph is one obtained by identifying two vertices (vertex identification); if the vertices are connected, this is called edge contraction.
