A Backtrack Procedure for Isomorphism of Directed Graphs

Abstract

A reasonably efficient procedure for testing pairs of directed graphs for isomorphism is important in information retrieval and other application fields in which structured data have to be matched. One such procedure, a backtrack procedure based on a representation of directed graphs by linear formulas, is described. A procedure for finding a partial subdigraph of a digraph that is isomorphic to a given digraph is also described.KEY WORDS AND PHRASES: backtrack programming, digraph isomorphism, representation of digraphs CR CATEGOnI~S: 5.32 DefinitionsA directed graph or digraph is the ordered pair D = (A, P}, where A is a set and P is a relation in A. We shall assume that A is finite, and put A = {aj, a2, • • •, a~}. Members of A are cMled nodes; members of P are arcs of the digraph. Arc (a~, aj} is said to originate from node a~ and to terminate at node ai. An are (ai, al} is called a sling. Digraph Dis completely specified by its adjace~cy matrix X, which is a square matrix of order n, defined as follows: xlj = 1 if ~al,as} E P, x~i= 0 if (at,as} ¢~ P.A digraph is total if every element of its adjacency matrix is 1.In digraph (A, P}, if certain members of P can be placed in a sequence (al, as}, (as, ak}, • • • , (as, at}, in which the first coordinate of each ordered pair is equal to the second coordinate of its predecessor in the sequence, then at is reachable from a~, and the set of arcs in the sequence is a path from a~ to at. Write the path as a node sequence (a~, as, ak, • .. , as, at). If this node sequence contains every node of the digraph exactly once, then the path is Hamiltonian; addition of arc tat, a~} to the Hamiltonian path gives a Hamiltonian cycle. We assume that every node is reachable from itself. If no arc originates from or terminates at a~, then a~ is an isolated node. The smallest set of nodes from which all nodes of a digraph are reachable is a node base of the digraph. A node base need not be unique, but all node bases of the one digraph contain the same number of nodes.If D = (A, P} is a digraph, and B is a subset of A, then D' = (B, (B X B) A P} is a subdigraph of D. If Q is a subset of P, .then D" = (A, Q} is a partial digraph of D.The concept of a partial subdigraph is an obvious extension.Two digraphs are isomorphic if some permutation of the rows and corresponding

An Algorithm for Subgraph Isomorphism

“…The experiments with cliques were carried out at the suggestion of Dr. B. R. Heap, who also proposed the modifications described in Appendix 2. The author is grateful to the referee for his comments and, in particular, for recommending comparison with References [2] and [3].…”

“…Like the tree-search algorithm of Berztiss [2], which we will discuss later, our algorithm differs from that of Corneil and Gotlieb in that it does not process the two graphs separately: the ability to cope with subgraph isomorphism stems from the fact that the computation always works on both graphs at once. We have not attempted to find a graph property that is possessed by all graphs which are isomorphic to a given graph.…”

“…DIGRAPH ISOMORPHISM. Berztiss [2] has provided a graph and subgraph isomorphism procedure for directed graphs. A directed graph or digraph consists of a finite nonempty set V of points, tegether with a set of ordered pairs of points in V. The algorithm introduced in the present paper is similar to that of Berztiss in that it works by tree searching instead of vertex classification, and it is an algorithm, not merely a heuristic procedure, and not based on an unproven conjecture.…”

Fast-On*: An Extended Algorithm for Graph Isomorphism Problem and Graph Query Processing

“…The first real dataset, referred to as AIDS_10K, consists of 10,000 graphs that are randomly drawn from the AIDS Antiviral screen database 1 . These graphs have 25 vertices and 27 edges on average.…”

“…In other words, it asks whether there is a one-to-one mapping between the vertices of the graphs, preserving the edges. This problem has been studied for decades by mathematicians, chemists and computer scientists, and is considered interesting from both the theoretical and the practical point of view, since it has applications in many fields, ranging from pattern recognition and computer vision [7] to information retrieval [1], data mining [19], or chemistry [14]. For example, in data mining, one main challenges in frequent subgraph mining is to systematically generate candidate subgraphs in a non-redundant manner, such that we do not generate the same graph more than once.…”