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Topic: Subgraph isomorphism problem

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	Graph isomorphism problem - Definition, explanation
		Besides its importance for solving a variety of practical problems, the graph isomorphism problem is a curiosity in complexity theory for defying the typical classifications that apply to other interesting practical problems.
		The complement of the graph isomorphism problem, sometimes called the graph nonisomorphism problem, is in co-NP, and was one of the first problems shown to be solvable by interactive proof systems, as well as one of the first problems for which a zero-knowledge proof was exhibited.
		The class GI Because the graph isomorphism problem is neither complete for any known classical class nor efficiently solvable, researchers sought to gain insight into the problem by defining a new class GI, the set of problems with a polynomial-time Turing reduction to the graph isomorphism problem.
	www.calsky.com /lexikon/en/txt/g/gr/graph_isomorphism_problem.php (698 words)

	Subgraph isomorphism problem - Wikipedia, the free encyclopedia
		In complexity theory, Subgraph-Isomorphism is a decision problem that is known to be NP-complete.
		The formal description of the decision problem is as follows.
		Subgraph isomorphism is a generalization of the potentially easier graph isomorphism problem; if this problem is NP-complete, the polynomial hierarchy collapses, so it is suspected not to be.
	en.wikipedia.org /wiki/Subgraph_isomorphism (169 words)

	Graph isomorphism problem - Wikipedia, the free encyclopedia
		Also, a generalization of the problem, the subgraph isomorphism problem, is known to be NP-complete.
		The complement of the graph isomorphism problem, sometimes called the graph nonisomorphism problem, is in co-NP, and was one of the first problems shown to be solvable by interactive proof systems, as well as one of the first problems for which a zero-knowledge proof was exhibited.
		The class GI Because the graph isomorphism problem is neither complete for any known classical class nor efficiently solvable, researchers sought to gain insight into the problem by defining a new class GI, the set of problems with a polynomial-time Turing reduction to the graph isomorphism problem.
	en.wikipedia.org /wiki/Isomorphism_problem (687 words)

	Professor Fred DePiero
		Subgraph isomorphism is a condition of isomorphism that exists between two subgraphs of G and H.
		However, the class of difficulty for the graph isomorphism problem is unknown [West01].
		It is due to the NP-Completeness of the subgraph isomorphism problem that an approximate solution was sought in the LeRP algorithm.
	www.ee.calpoly.edu /~fdepiero/fdepiero_research/subgraph_intro.html (986 words)

	Graph theory - Wikipedia, the free encyclopedia
		In 1852 Francis Guthrie posed the four color problem which asks if it is possible to color, using only four colors, any map of countries in such a way as to prevent two bordering countries from having the same color.
		This problem, which was only solved a century later in 1976 by Kenneth Appel and Wolfgang Haken, can be considered the birth of graph theory.
		Covering problems are specific instances of subgraph-finding problems, and they tend to be closely related to the clique problem or the independent set problem.
	en.wikipedia.org /wiki/Graph_theory (1782 words)

	[No title]
		Because the largest common subgraph problem is a polynomial problem for trees, LCSA explores the spanning trees of the graphs and uses a backtracking method to be exhaustive in its search.
		Subsequently, subgraph isomorphism is considered as a special case of graph similarity and a new efficient algorithm for its detection is proposed.
		We focus on the problem of partitioning a partial $k$-tree into induced subgraphs isomorphic to a fixed {\em pattern} graph; a distinct algorithm is derived for each pattern graph and each value of $k$.
	www.ics.uci.edu /~eppstein/bibs/subiso.bib (13928 words)

	Clique (graph theory) - Wikipedia, the free encyclopedia
		In graph theory, a clique in an undirected graph G, is a set of vertices V such that for every two vertices in V, there exists an edge connecting the two.
		This is equivalent to saying that the subgraph induced by V is a complete graph.
		The opposite of a clique is an independent set, in the sense that every clique corresponds to an independent set in the complement graph.
	en.wikipedia.org /wiki/Clique_(graph_theory) (167 words)

	Subgraph Isomorphism, log-Bounded Fragmentation, and Graphs of (Locally) Bounded Treewidth (ResearchIndex) (Site not responding. Last check: )
		Abstract: The subgraph isomorphism problem, that of nding a copy of one graph in another, has proved to be intractable except when certain restrictions are placed on the inputs.
		In this paper, we introduce a new property for graphs along with an associated graph class (a generalization on bounded degree graphs) and extend the known classes of inputs for which polynomial-time subgraph isomorphism algorithms are attainable.
		6.9%: Subgraph Isomorphism, log-Bounded Fragmentation and Graphs of..
	citeseer.ifi.unizh.ch /681084.html (548 words)

	Faster Algorithms for Subgraph Isomorphism of k-connected Partial k-trees - Dessmark, Lingas, Proskurowski ... (Site not responding. Last check: )
		Abstract: The problem of determining whether a k-connected partial k-tree is isomorphic to subgraph of another partial k-tree is shown to be solvable in time O(n k+2).
		1 Introduction The subgraph isomorphism problem is to determine whether a graph is isomorphic to a subgraph of another graph.
		1 A Polynomial Algorithm for Subgraph Isomorphism of Two-conne..
	citeseer.ifi.unizh.ch /281990.html (575 words)

	GRAPH ISOMORPHISM PROBLEM: AN ALGORITHM FOR SOLUTION
		The graph isomorphism problem can not be placed in any of known complexity classes as it was stated in [1].
		The isomorphism of the graphs checks at most at n iterations of the algorithm, where n is a number of vertices of graphs.
		It is shown that this is holds regardless of maximal degree of the graphs, graphs genus, graph eigenvalue multiplicity etc., i.e., using the algorithm, solution of the graph isomorphism problem has no specific that may be determinate by any graph characteristics that is usually considered.
	www.psy.omsu.omskreg.ru /session/isomorphism (501 words)

	Gemini
		While the subgraph isomorphism problem is known to be NP-complete in general, the SubGemini algorithm can efficiently solve this problem for most circuits encountered in practice by making use of their inherent structure.
		We have approached the general problem by treating it as an instance of the subgraph isomorphism problem.
		While the subgraph isomorphism problem is known to be NP-complete in general, the algorithm presented in this paper can efficiently solve this problem for most circuits encountered in practice by making use of the structure found in circuits.
	www.cs.washington.edu /research/projects/lis/www/gemini/gemini.html (852 words)

	Algorithmic aspects of classes with bounded expansion
		It appeared that many NP-complete problems may be solved in polynomial time when restricted to a class with bounded tree-width.
		It is quite natural to ask whether the parts could be choosen "smaller" or "simpler" This aspects has been studied in [13] where the tree-depth td(G) of a graph G is introduced as the minimal height of a rooted forest whose closure includes G.
		Subgraph isomorphism in planar graphs and related problems.
	www.ehess.fr /centres/cams/person/pom/langen/grad.html (710 words)

	isomorphism Isomorphism - Wikipedia, the free encyclopedia
		Isomorphism -- From MathWorld Informally, an isomorphism is a map that preserves sets and relations among elements.
		Isomorphism Up to isomorphism, there is only one group with a prime number of elements.
		Isomorphism The applicability of a theory is explained by examining its isomorphism with a model of the phenomena it Introduction.
	adenosis.blog9.businessweekblog.com /1143328481.html (605 words)

	[No title] (Site not responding. Last check: )
		Solution to in-class exercise: In the in-class exercise, you were asked to show that the subgraph isomorphism problem is NP-HARD by giving a transformation of the k-Clique problem.
		The input to the k-Clique problem is a graph G and an integer k; the subgraph isomorphism problem takes for input two graphs G1 and G2.
		Now suppose (for the sake of argument) that we were able to develop a polynomial time algorithm solving the subgraph isomorphism problem.
	www.cs.siena.edu /~flatland/csis385s99_hmwk6sol.html (245 words)

	Citations: Linear Time Algorithms for Isomorphisms of Planar Graphs - Hopcroft, Wong (ResearchIndex)
		In general, the problem of graph isomorphism (step (2) is not known to be either in the P or NP complete class (see [26, p.
		The planar graph isomorphism is a polynomially solvable problem.
		Notice that this latter problem is not known to be either polynomial or NP complete in general.
	citeseer.ist.psu.edu /context/276752/0 (2322 words)

	NP-complete (Site not responding. Last check: )
		One example of an NP-complete problem is the subset sum problem which is: given a finite set of integers, determine whether any nonempty subset of them adds up to zero.
		The Graph Isomorphism problem is suspected to be neither in P nor NP-complete, though it is obviously in NP.
		A problem X is polynomial time, Turing reducible to a problem Y if, given a subroutine that solves Y in polynomial time, you could write a program that calls this subroutine and solves X in polynomial time.
	www.termsdefined.net /np/np-complete.html (1161 words)

	[No title] (Site not responding. Last check: )
		Problems involving the coloring of graphs are notoriously intractable.
		Given the intractability of many problems in graph theory, it is natural that this area has given rise to the development of many approximation algorithms.
		Unless there are polynomial-time solutions for the NP-complete problems (an unlikely possibility), it has been proved that there can be no polynomial-time solution that gives a vertex coloring of an arbitrary graph that guarantees to use less than twice the minimum number of colors required.
	cs.union.edu /~hemmendd/Encyc/Articles/Graphtheory/graph.art.html (2237 words)

	Citations: Subgraph isomorphism in planar graphs and related problems - Eppstein (ResearchIndex)
		The subgraph isomorphism problem for the source graph G and the host graph H is NP complete.
		To solve subproblem (2) we transform the problem into the well known set covering problem (Chvatal 1979) In the set covering problem, the data consists of finite sets P 1,P 2, P n and positive numbers c 1,c 2, c n.
		Basically, a planar graph is one which can be drawn on a plane in such a way that there are no crossing edges (thus, for instance, the graph in Figure 2 is planar) It is worth investigating to what extent planar graphs suffice for the generation of referring expressions.
	citeseer.ist.psu.edu /context/281922/240931 (2854 words)

	Introduction to Algorithms \| Glossary
		Informally, a problem is in the class NPC - and we refer to it as being NP-complete - if it is in NP and is as "hard" as any problem in NP.
		of an abstract problem are polynomially related, whether the problem is polynomial-time solvable or not is independent of which encoding we use.
		Modeling the problem as a complete graph with n vertices, we can say that the salesman wishes to make a tour, or hamiltonian cycle, visiting each city exactly once and finishing at the city he starts from.
	highered.mcgraw-hill.com /sites/0070131511/student_view0/chapter34/glossary.html (2111 words)

	DNA Computing - Overview, Articles and Links
		The problem is deceptively easy to describe, but in fact belongs to the notoriously intractable class of NP-complete problems, which signifies the class of problems solvable in Nondeterministic Polynomial(NP) time.
		Typically, these problems involve a search where at each point in the search there is an exponential increase in the number of possibilities to be searched through, but where each possibility can be searched through polynomial time.
		It is sufficiently strong to solve any of the problems in the class NC and the authors have given DNA algorithms for 3-vertex-colorability problem, Permutations Problem, Hamiltonian Path Problem, the Subgraph isomorphism problem, and the Maximum clique and maximum independent set problem.
	www.peterindia.net /DNAComputing.html (1957 words)

	CS345: Project on graph isomorphism
		The subgraph isomorphism problem is a well known problem for algorithm engineers.
		It is, given a graph G and a graph g, to verify whether "G has a subgraph which is isomorphic to g" or not.
		So, the goal of this project is to come up with some heuristic approximation(polynomial) algorithm for the problem of subgraph isomorphism, which runs with high probabilty of success.
	www.cse.iitk.ac.in /users/dsrkg/cs245/html/proj2.htm (139 words)

	NP
		The reduction is a polynomial-time computable function that maps each instance of the chosen NP-complete problem into an instance of PROB in a way that the original instance is a "yes"-instance of the chosen NP-complete problem iff the mapped instance is a "yes"-instance of PROB.
		Notice that, while SUBGRAPH ISOMORPHISM is NP-complete, the problem GRAPH ISOMORPHISM asking whether two graphs are isomorphic is not known to be NP-complete nor in P (it certainly is in NP because we can guess the isomorphism mapping non-deterministically and then verify it in deterministic polynomial time).
		The decision problem of longest problem is Is there a simple path in G (between two arbitrary distinct vertices) of length at least k.
	www.ecst.csuchico.edu /~amk/foo/csci356/notes/ch11/HW11.html (2036 words)

	CSCI 4602 solutions to practice questions for quiz 5 (Site not responding. Last check: )
		Decision problem A is NP-complete if A is in NP and for every X in NP, X reduces to A in polynomial time.
		Is H isomorphic to a subgraph of G? That is, can you remove zero or more vertices and zero or more edges from G and obtain a graph that has the same structure as H? Show that the clique problem polynomial-time reduces to the subgraph isomorphism problem.
		The problem of testing whether two given graphs are isomorphic is not known to be solvable in polynomial time (and is conjectured not to be solvable in polynomial time).
	www.cs.ecu.edu /~karl/4602/fall01/pracsoln5.html (625 words)

	[No title]
		Title: Subgraph Isomorphism in Planar Graphs and Related Problems
		We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size.
		The same methods can be used to solve other planar graph problems including diameter, girth, induced subgraph isomorphism, and shortest paths.
	www.netlib.org /confdb/soda95/Epps.html (88 words)

	Citebase - Subgraph Isomorphism in Planar Graphs and Related Problems
		Subgraph Isomorphism in Planar Graphs and Related Problems
		We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size.
		The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:cs/9911003 (147 words)

	CSCI 6420, Spring 2002
		TIME(t(n)) is the class of all decision problems that are solvable in time at most c*t(n) for some constant c on inputs of length n, for all but finitely many n.
		We defined the class P of all decision problems solvable in polynomial time, and saw some examples of problems in P. Material on polynomial time computation is in Chapter 7 of the text.
		If P is not equal to NP and problem B is NP-complete then B cannot be in P. It is conjectured (but not known) that P is not equal to NP.
	www.cs.ecu.edu /~karl/6420/spr02 (975 words)

	[No title]
		NNTP-Posting-Host: diego.llnl.gov Keywords: MCS, isomorphism, graphs, NP-Complete Originator: cedeno@diego.llnl.gov Following is the summary on the responses I received on algorithms for finding the common subgraphs between two graphs.
		The latter allows you to test isomorphism, but there is no built-in support for subgraph isomorphism.
		You can approximate the largest subgraph to a factor, meaning that you are not guaranteed to find the largest graph, but you'll find a neat approximation.
	ai.stanford.edu /~suresh/theory/references/MCS.txt (674 words)

	Subgraph isomorphism problem - Definition, explanation
		In complexity theory, Subgraph-Isomorphism is a decision problem that is known to be NP-complete.
		This name puts emphasis on finding such a subgraph and is not a bare decision problem.
		Subgraph isomorphism is a generalization of the potentially easier graph isomorphism problem.
	www.calsky.com /lexikon/en/txt/s/su/subgraph_isomorphism_problem.php (149 words)

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