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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
157:) 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.
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A directed graph (blue and black) and its condensation (yellow). The strongly connected components (subsets of blue vertices within each yellow vertex) form the blocks of a partition giving rise to the
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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.
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under the covering map. However, covering maps have an additional requirement that is not true more generally of quotients, that the map be a local isomorphism.
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134:– so its objects can be regarded as "sets with additional structure", and a quotient graph corresponds to the graph induced on the
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Faria, L.; de
Figueiredo, C. M. H.; Mendonça, C. F. X. (2001), "Splitting number is NP-complete",
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355:; Somenzi, Fabio (January 2006), "An algorithm for strongly connected component analysis in
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249:. The blocks of the corresponding partition are the inverse images of the vertices of
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324:, Contemp. Math., vol. 588, Amer. Math. Soc., Providence, RI, pp. 1–17,
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form the blocks of the partition. This construction can be used to derive a
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induced by the partition, then the quotient graph has vertex set
318:; Schulz, Christian (2013), "High quality graph partitioning",
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of a directed graph is the quotient graph where the
130:– mapping a graph to its set of vertices makes it a
398:Gardiner, A. (1974), "Antipodal covering graphs",
170:); if the vertices are connected, this is called
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31:is a graph whose vertices are blocks of a
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321:Graph partitioning and graph clustering
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166:obtained by identifying two vertices (
126:of graphs. The category of graphs is
118:More formally, a quotient graph is a
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284:can be obtained as a quotient of a
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55:with respect to the edge set of
400:Journal of Combinatorial Theory
365:Formal Methods in System Design
218:, in which the blocks are the
51:is adjacent to some vertex in
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451:10.1016/S0166-218X(00)00220-1
197:strongly connected components
438:Discrete Applied Mathematics
413:10.1016/0095-8956(74)90072-0
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206:The result of one or more
377:10.1007/s10703-006-4341-z
203:from any directed graph.
178:Special types of quotient
257:Computational complexity
280:, to determine whether
245:is a quotient graph of
210:in an undirected graph
330:10.1090/conm/588/11700
201:directed acyclic graph
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149:. Further, there is a
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168:vertex identification
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43:is adjacent to block
220:connected components
77:equivalence relation
222:of the subgraph of
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151:graph homomorphism
145:of its vertex set
47:if some vertex in
363:symbolic steps",
351:Bloem, Roderick;
237:of another graph
214:is a quotient of
208:edge contractions
132:concrete category
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484:Quotient objects
479:Graph operations
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353:Gabow, Harold N.
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286:planar graph
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473:Categories
302:References
298:vertices.
187:quotient.
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