In particular, notice that the result of this process is a planar graph. We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Algorithm and experiments in testing planar graphs for isomorphism. It is known to be npcomplete, even when restricted to planar graphs 18. Maximum common subgraph isomorphism algorithms for the matching of chemical. Note that subgemini, which is a 1993 circuitnetlistoriented subgraph isomorphism solver, doesnt use a planar algorithm, seemingly because they did not want to make planarity assumptions for subgraph isomorphism in general i. Subgraph isomorphism for biconnected outerplanar graphs in. Although polynomialtime isomorphism algorithms are known for various graph classes, like trees and planar graphs 5. Schnorr 8 have presented a convex programming approach for subgraph matching. Subgraph isomorphism is a generalization of both the maximum clique problem and the problem of testing whether a graph contains a hamiltonian cycle, and is therefore npcomplete.
Our results are based on a technique of partitioning the planar graph into pieces of small treewidth, and applying dynamic programming within each piece. Pdf the graph isomorphism problem restricted to planar graphs has been known to be solvable in polynomial time many years ago. Subgraph isomorphism on boundedtreewidth graphs is due to matou sek and thomas 50. However, this problem is npcomplete garey and johnson, 1979 and the problem of. For both subgraph isomorphism and maximum common subgraph, constraint programming is the best known approach1, although a reduction to the maximum clique problem is better when edge labels are present ndiaye and solnon 2011. Some constraints needs to be evolved in order to solve subgraph isomorphism in polynomial time even if it is only for some restricted class of graphs. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths. Enumerating all subgraphs of a given graph takes exponential time. In this paper, we proposeanew algorithm which can deal with this problem. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph.
And almost the subgraph isomorphism problem is np complete. Planar subgraph isomorphism revisited internet archive. Im not aware of any implementations of planar subgraph isomorphism algorithms, sorry. Subgraph isomorphism and its application to layoutversus. An indepth comparison of subgraph isomorphism algorithms in graph databases. Uses an idea of baker to cover a planar graph with subgraphs of low treewidth. Induced subgraph isomorphism is a special case let k order of pattern graph. Lingas, subgraph isomorphism for easily separable graphs of bounded valence, 11th workshop on graphtheoretic concepts in computer. Algorithm and experiments in testing planar graphs for.
In this paper we consider planar graph isomorphism and settle its complexity by. In this section, we study the subgraph isomorphism problem on patterns and host graphs that are embedded in a sphere in s ection 5 we carry o ver our results to planar graphs. Planar and nonplanar graphs, and kuratowskis theorem. An algorithm for maximum common subgraph of planar. A simple non planar graph with minimum number of vertices is the complete graph k5. The simple non planar graph with minimum number of edges is k3, 3. A subgraph of g is a graph whose points and lines all belong to g.
Planar graphs a graph g is said to be planar if it can be drawn on a plane or a sphere so that no two edges cross. Thus mcis is nphard kclique is another special case. An optimization of closed frequent subgraph mining. The subgraph isomorphism problem for twoconnected outerplanar graphs can be sequentially solved by a recursive reduction to a monotone path finding problem in cubic time 14. Department of information and computer science university of california, irvine, ca 92717 tech. Subgraph and induced subgraph isomorphism problems are known to be npcomplete 4, while the graph isomorphism problem is one of the few problems in np neither known to be in p nor npcomplete. Until now, the best known algorithm to solve subgraph isomorphism, that is to. Subgraph isomorphismgeneralizes many important graph problems, such as hamiltonicity, longest path, and clique.
We know that a graph cannot be planar if it contains a kuratowski subgraph, as those subgraphs are nonplanar. Subgraph isomorphism and its application to layoutversusschematic veri cation nathan breitsch march 28, 2016. Trees and partial 2trees are special cases of planar graphs. It is at least related you probably already know it. F or the sp ecial case of triv alen t graph isomorphism, it w as sho in luk82 that algorithms with a computational complexit y of o n 6 exist. A graph has locally boundedtreewidthif the treewidth of the subgraph induced on all vertices at distance r. The proposed algorithm can solve the subgraph isomorphism in polynomial time in some settings.
Homomorphism two graphs g 1 and g 2 are said to be homomorphic, if each of these graphs can be obtained from the same graph g by dividing some edges of g with more vertices. In hw74 metho d for the computation of isomorphism planar graphs is prop osed that has only a linear time complexit y. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced. Mcis remains nphard on most known graph classes including bipartite graphs, planar graphs, and graphs of bounded treewidth. Subgraph isomorphism in planar graphs and related problems. March 4 if you have not yet turned in the problem set, you should not consult these solutions. For k 2 and we also show that lsubgraph contractibility remains npcomplete when we contract to graphs of higher genus g instead of planar graphs. The algorithm requires time, if v is the number of vertices in each graph. Pdf the complexity of planar graph isomorphism researchgate. First, observe that subgroup isomorphism is in np, because if we are given a speci cation of the subgraph of g and the mapping between its vertices and the vertices of. The induced subgraph isomorphism computational problem is, given h and g, determine whether there is a induced subgraph isomorphism from h to g. The subgraph isomorphism also remains npcomplete when the first graph is a forest and the other input graph is a tree see p. Let g be a planar graph on n vertices and h a pattern of.
Given two graphs g,h on n vertices distinguish the case that they are isomorphic from the case that they are not isomorphic is very hard. Strongly connected component analogous to connected components in undirected graphs, a strongly connected component is a subgraph of a. Baker bak94, using a decomposition of a planar graph into outerplanar graphs, found ecient approximation algorithms for planar graphs section 2. Planar triangulation graphs, which are commonly used. The subgraph isomorphism problem was also studied on graphs with topological restrictions. If such an f exists, then we call fh a copy of h in g. 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 search is the basic type of graph queries. Planar graph isomorphism appeared especially interesting after hopcroft and wong. A graph g, is somorphw to a subgraph of a graph ga if and only if there is a 1.
Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Theoretical research 28 indicates a linear time complexity for testing planar. Planar subgraph isomorphism revisited 265 in the size of the host graph, we give a fast method for computing spherecut decompositionsnatural extensions of treedecompositions to plane graphs with separators of size linearly bounded by the size of the subgraph pattern. Weinberg wei66 presented an on2 algorithm for testing isomorphism of 3connected planar graphs. Thus, the subgraph isomorphism problem is npcomplete even if g and h range only over connected planar graphs of valence s3. In the graph g3, vertex w has only degree 3, whereas all the other graph vertices has degree 2. Report 9425 may 31, 1994 abstract we solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. This article focuses on degeneracy of planar graphs. In this paper, we focus on a special class of graphs, i. A graph g is non planar if and only if g has a subgraph which is homeomorphic to k 5 or k 3,3. Subgraph isomorphism, logbounded fragmentation, and graphs of. The problem subgraph isomorphism is a fundamental problem in graph theory. The problem of subgraph isomorphism is defined as follows. The best algorithm is known today to solve the problem has run time for graphs with n vertices.
Largest common subgraph of two maximal planar graphs. For connected h, subgraph isomorphism can be solved in time f h notwg for some computable function f. Given a query graph q and a data graph g, the subgraph isomorphism problem is to. Planar graph isomorphism turns out to be complete for a wellknown and natural complexity class, namely logspace. As stated above, our goal is to prove that these necessary condi. A subgraph isomorphism algorithm for matching large graphs. Polynomial algorithms for open plane graph and subgraph.
Subgraph isomorphism in planar graphs and related problems david eppstein. A graph h v h, e h is subgraphisomorphic to a graph g v g, e g if there exists an injective map. Subgraph isomorphism in graph classes sciencedirect. We propose a new fast algorithm for solving the maximum common subgraph mcs problem. If h is part of the input, subgraph isomorphism is an npcomplete problem.
The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and. Planar graph isomorphism has been studied in its own right since the early days of computer science. Given a pattern h and a host graph g on n vertices, does g contain a. Normally, the subgraph isomorphism problem is npcomplete. Subgraph isomorphism, logbounded fragmentation, and. Subgraph isomorphism search in massive graph databases. Subgraph isomorphism is npcomplete when the source graph is a tree and the host graph is a partial 2tree that has at most one. The graph is weakly connected if the underlying undirected graph is connected.