Graph matching is the problem of finding a similarity between

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

s.Endika Bengoetxea
"Inexact Graph Matching Using Estimation of Distribution Algorithms"
, Ph. D., 2002
Chapter 2:The graph matching problem
(retrieved June 28, 2017) Graphs are commonly used to encode structural information in many fields, including

computer vision Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical ...

and

pattern recognition Pattern recognition is the task of assigning a class to an observation based on patterns extracted from data. While similar, pattern recognition (PR) is not to be confused with pattern machines (PM) which may possess PR capabilities but their p ...

, and graph matching is an important tool in these areas.Endika Bengoetxea, Ph.D.
Abstract
In these areas it is commonly assumed that the comparison is between the ''data graph'' and the ''model graph''. The case of exact graph matching is known as the

graph isomorphism problem The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational c ...

. The problem of exact matching of a graph to a part of another graph is called

subgraph isomorphism problem In theoretical computer science, the subgraph isomorphism problem is a computational task in which two graphs G and H are given as input, and one must determine whether G contains a subgraph that is isomorphic to H. Subgraph isomorphism is a gen ...

. Inexact graph matching refers to matching problems when exact matching is impossible, e.g., when the number of vertices in the two graphs are different. In this case it is required to find the best possible match. For example, in

image recognition Computer vision tasks include methods for acquiring, processing, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical or symbolic information, e.g. in the form o ...

applications, the results of

image segmentation In digital image processing and computer vision, image segmentation is the process of partitioning a digital image into multiple image segments, also known as image regions or image objects (Set (mathematics), sets of pixels). The goal of segmen ...

image processing An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ...

typically produces data graphs with the numbers of vertices much larger than in the model graphs data expected to match against. In the case of

attributed graph In computer science, an attributed graph grammar is a class of graph grammar that associates vertices with a set of attributes and rewrites with functions on attributes. In the algebraic approach to graph grammars, they are usually formulated using ...

s, even if the numbers of vertices and edges are the same, the matching still may be only inexact. Two categories of search methods are the ones based on identification of possible and impossible pairings of vertices between the two graphs and methods that formulate graph matching as an

optimization problem In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goo ...

. Graph edit distance is one of

similarity measure In statistics and related fields, a similarity measure or similarity function or similarity metric is a real-valued function that quantifies the similarity between two objects. Although no single definition of a similarity exists, usually such mea ...

s suggested for graph matching.Horst Bunke, Xiaoyi Jang, "Graph Matching and Similarity", in: ''Intelligent Systems and Interfaces'', pp. 281-304 (2000) The class of algorithms is called error-tolerant graph matching.

References

Computational problems in graph theory {{Comp-sci-stub

See also

References