A Graph Isomorphism Filter for the GP Language Philip Cavanagh
Graph Isomorphisms • Isomorphism: Two graphs which are “the same” up to a renaming of the vertices and edges • These are a pair of isomorphic graphs:
Graph Isomorphisms • Isomorphism: Two graphs which are “the same” up to a renaming of the vertices and edges • These are a pair of isomorphic graphs:
The Graph Isomorphism Problem • A decision problem which asks, “Given two graphs, G and H , is G isomorphic to H ?” • In NP, as a map from G to H can be checked in polynomial time by comparing adjacency matrices • Not known to be in either P or NP-Complete • Can be considered its own class between the two: GI
What is GP? • Graph Programming Language • Applies graph transformation rules • Rules are applied in a non-deterministic manner as defined by the program text • Transforms a start graph into a set of output graphs
GP Example: Program Rule name: link Program Text: “main = link!.”
GP Example: Rule Application link
GP Example: Output Graphs
Isomorphism Filter • Removal of isomorphic copies in this output set would be beneficial • An adapted graph isomorphism problem: • Given a set of graphs S, return the smallest subset S’ of S such that every graph in S is isomorphic to a graph in S’ • This “filters out” unwanted duplicates
Current Approaches • Direct Mapping – Can “stop early” if isomorphism found – Can only compare two graphs • Canonical Forms (e.g. lexicographically smallest adjacency matrix) – Can compare many graphs – Requires full search to give an answer
A Middle Ground? • José Luis López Presa, 2009 Thesis • Takes early stopping from direct mapping • Can be extended to compare many graphs • Seems a good method for the problem
Presa’s Algorithm (Graph Pair) • Generate an ordering on the vertices of the first graph • Try to generate a compatible ordering on the second graph • The two orderings can be compared to see if they define an isomorphism
Example • We use the following graphs to compare for isomorphism. First, graph G needs to be processed.
Generating a Vertex Ordering • Generate an initial partition according to the vertices’ degrees within the graph • All vertices in G have degree 3, so the initial partition is P 0 = ({0,1,2,3,4,5,6,7}) • Repeatedly refine the partition using (in order of preference): – Vertex refinement – Set refinement – Backtrack refinement
Backtrack Refinement • As we only have one partition cell, we start with a backtrack refinement • A vertex is chosen from the cell (e.g. 1) • Any vertex adjacent to 1 is placed in one cell • The others are placed in a separate cell • P 1 = ({0,2,7},{3,4,5,6})
Set Refinement • Now that we have two cells, we can try to split them based on their adjacencies with each other. • P 1 = ({0,2,7},{3,4,5,6}) • We use cell 1 to try to split both of the cells of P 1 • P 2 = ({2,7},{0},{6},{3,4},{5})
Vertex Refinement • We now use the third type of refinement as we have a cell containing only one vertex • This divides cells into those vertices which are and are not adjacent to the chosen vertex • P 2 = ({2,7},{0},{6},{3,4},{5}) • We choose a vertex (e.g. 0) and split the other cells with it • P 3 = ({2,7},{6},{4},{3},{5})
Vertex Refinement (II) • P 3 = ({2,7},{6},{4},{3},{5}) • We can use vertex refinement again, this time using vertex 6: • P 4 = ({7},{2},{4},{3},{5})
Obtaining the ordering for G • We now have a partition of single vertex cells: P 4 = ({7},{2},{4},{3},{5}) • We can now obtain an ordering by considering the vertices removed (1, 0 and 6) and then adding the final partition: • Order( G ) = (1,0,6,7,2,4,3,5)
Now for H • To find an isomorphism between G and H , we need to be able to generate a similar looking partition for H as for G using the same refinement types at each stage. • Again, all the vertex degrees are 3, so we start with a partition Q 0 = ({0,1,2,3,4,5,6,7})
First Attempt • As with G , we need to start with a backtrack refinement. This time we pick vertex 0. • Then Q 1 = ({1,2,7},{3,4,5,6}) • P 1 was ({0,2,7},{3,4,5,6}), so we can continue • A set refinement with cell 1 gives Q 2 = ({1,2},{7},{3,4,5,6})
Backtracking • Q 2 = ({1,2},{7},{3,4,5,6}), while P 2 = ({2,7},{0},{6},{3,4},{5}) • These are not compatible, so we are forced back to the last backtracking refinement to try a different vertex. • Pick vertex 1 for backtracking: • Q 1 = ({0,2,4},{3,5,6,7})
Continuing • By applying a set refinement by cell 1, then vertex refinements by vertices 4 and 3, we get the following, which are compatible with P 1 to P 4 • Q 1 = ({0,2,4},{3,5,6,7}) • Q 2 = ({0,2},{4},{3},{5,7},{6}) • Q 3 = ({0,2},{3},{5},{7},{6}) • Q 4 = ({2},{0},{5},{7},{6})
Comparing orderings • We now have a pair of orderings for G and H : • Order( G ) = (1,0,6,7,2,4,3,5) • Order( H ) = (1,4,3,2,0,5,7,6) • We can see this represents an isomorphism
Extension to GP Graphs • Graphs in GP are more general than this example. Specifically, GP allows for graphs to have: – Vertex labels and loops: can be taken into consideration in the initial degree partition – Edge labels, directed edges and parallel edges: a slightly more complex definition of the degree of a vertex allows for this
Extension to Many Graphs • Store successive refinements of a graph’s degree partition in a tree structure • When testing a new graph, try to match its partition refinements to existing branches (only possible at backtracking nodes) • If no branch can be matched, the graph is not isomorphic and should be added to the tree down a new set of branches
Summary • GP can have a large number of duplicated graphs in the output set • Presa’s algorithm compares pairs of graphs, taking good points from direct and canonical form methods • An adaptation of this algorithm can reduce GP’s output set to a smaller set which is simpler for the user
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