Quotient graph
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)}. For example, the condensation of a directed graph is the quotient graph where the strongly connected components form the blocks of the partition.[2]
The result of one or more edge contractions in an undirected graph G is a quotient of G, in which the blocks are the connected components of the subgraph of G formed by the contracted edges. However, for quotients more generally, the blocks of the partition giving rise to the quotient do not need to form connected subgraphs.
References
- ^ Sanders, Peter; Schulz, Christian (2013), "High quality graph partitioning", Graph partitioning and graph clustering, Contemp. Math., vol. 588, Amer. Math. Soc., Providence, RI, pp. 1–17, doi:10.1090/conm/588/11700, MR 3074893.
- ^ Bloem, Roderick; Gabow, Harold N.; Somenzi, Fabio (January 2006), "An algorithm for strongly connected component analysis in n log n symbolic steps", Formal Methods in System Design, 28 (1): 37–56, doi:10.1007/s10703-006-4341-z.
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