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GraphIsomorphisms.hpp File Reference

Algorithms to determine graph isomorphisms (directly or by reducing it to (generalised) SAT problems) More...

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Detailed Description

Algorithms to determine graph isomorphisms (directly or by reducing it to (generalised) SAT problems)

Given two graphs G_1, G_2 together with maps v in V(G_1) -> D(v) <= V(G_2) specifying to which vertices v may me mapped, find a graph isomorphism from G_1 to G_2 fulfilling these specifications.

For direct graph isomorphism algorithms see Combinatorics/Graphs/Isomorphisms/plans/general.hpp.

Todo:
Vertex invariants
  • The first step consists in applying algorithms which further reduce the sets D(v), for example by using that isomorphisms maintain vertex degrees.
  • More generally, each vertex is labelled by some list of multisets of natural numbers, such that isomorphisms must maintain these lists; the first step is to use just the vertex degrees, the second step is to use for each vertex the multiset, which maps each possible vertex degree of a neighbour to the count of such neighbours; and so on (is this the idea of "nauty" ?).
  • Another possibility is to use adjacency relations: If w is adjacent to v, then for every value w' in D(w) there must exist a value v' in D(v) such that w' is adjacent to v'.
  • Either such algorithms are processed up to a certain level, or "until they get stuck", and then the remaining problem is formulated as a satisfiability problem.
Todo:
Alliances
  • Still just given G_1, G_2 and D, but D may have been further refined by the previous step.
  • We obtain a alliance of active clause-sets with variable set V(G_1) and variable domains given by D, with members
    1. BIJ(V(G_1), D, V(G_2)) (see module "Generators").
    2. A virtual clause-set which in principal for every vertex v and set V of possible values can yield for every vertex w adjacent to v in G the set of possible values W given by the set of w' such that v' in V with {v',w'} in E(G_2) exists.
    3. Of main importance are the cases with |V| = 1 (these conditions ensure completeness of the translation).
    4. This active clause-set may also offer stronger inferences.
    5. In principal it is again an algorithm from the first step, processing a refined domain-assignment as input and computing further refinements.
  • This alliance then is passed to a SAT algorithm (or output, using the canonical clause-representation).
Todo:
What about the graph embedding problem ? This should likely go to another module.

Definition in file GraphIsomorphisms.hpp.