the GP Language Philip Cavanagh Graph Isomorphisms Isomorphism: - - PowerPoint PPT Presentation

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 P0 = ({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

• P1 = ({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

• ther.
• P1 = ({0,2,7},{3,4,5,6})
• We use cell 1 to try to split

both of the cells of P1

• P2 = ({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

• P2 = ({2,7},{0},{6},{3,4},{5})
• We choose a vertex (e.g. 0)

and split the other cells with it

• P3 = ({2,7},{6},{4},{3},{5})

Vertex Refinement (II)

• P3 = ({2,7},{6},{4},{3},{5})
• We can use vertex refinement again, this time

using vertex 6:

• P4 = ({7},{2},{4},{3},{5})

Obtaining the ordering for G

• We now have a partition of single vertex cells:

P4 = ({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 Q0 = ({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 Q1 = ({1,2,7},{3,4,5,6})
• P1 was ({0,2,7},{3,4,5,6}), so we can continue
• A set refinement with cell 1

gives Q2 = ({1,2},{7},{3,4,5,6})

Backtracking

• Q2 = ({1,2},{7},{3,4,5,6}), while

P2 = ({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:
• Q1 = ({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 P1 to P4

• Q1 = ({0,2,4},{3,5,6,7})
• Q2 = ({0,2},{4},{3},{5,7},{6})
• Q3 = ({0,2},{3},{5},{7},{6})
• Q4 = ({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