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)}.

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.

Examples

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.

Special types of quotient
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 quotient.

The condensation of a directed graph is the quotient graph where the strongly connected components form the blocks of the partition. This construction can be used to derive a directed acyclic graph from any directed graph.[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.

If G is a covering graph of another graph H, then H is a quotient graph of G. The blocks of the corresponding partition are the inverse images of the vertices of H 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.[3]

Computational complexity

It is NP-complete, given an n-vertex cubic graph G and a parameter k, to determine whether G can be obtained as a quotient of a planar graph with n + k vertices.[4]

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References
  1. Sanders, Peter; Schulz, Christian (2013), "High quality graph partitioning", Graph partitioning and graph clustering, Contemp. Math., 588, Amer. Math. Soc., Providence, RI, pp. 1–17, doi:10.1090/conm/588/11700, MR 3074893.
  2. 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.
  3. Gardiner, A. (1974), "Antipodal covering graphs", Journal of Combinatorial Theory, Series B, 16: 255–273, doi:10.1016/0095-8956(74)90072-0, MR 0340090.
  4. Faria, L.; de Figueiredo, C. M. H.; Mendonça, C. F. X. (2001), "Splitting number is NP-complete", Discrete Applied Mathematics, 108 (1–2): 65–83, doi:10.1016/S0166-218X(00)00220-1, MR 1804713.
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