D. Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms & Applications, 3(3):1--27,1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(extended abstract) Erik D. Demaine and MohammadTaghi Hajiaghayi Laboratory for Computer Science, Massachusetts Institute of Technology, 200 Technology Square, Cambridge, MA 02139, U.S.A. fedemaine,hajiaghag theory.lcs.mit.edu Abstract. We solve an open problem posed by Eppstein in 1995 [16, 17] and re enforced by Grohe [18, 19] concerning locally bounded treewidth in minor closed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a function of r (not n) Eppstein characterized ....

....times of 2 for many problems on planar graphs, such as maximum independent set, minimum dominating set, and minimum vertex cover. Chen [10] later generalized Baker s approach to obtain PTASs for K 3;3 minor free graphs and K 5 minor free graphs, but only for maximization problems. Eppstein [17, 16] further generalized Baker s approach by replacing local regions of bounded outerplanarity with local regions of bounded treewidth. The resulting approximation algorithms apply to any graph of bounded local treewidth, a notion newly introduced by Eppstein [16, 17] A graph has bounded local ....

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David Eppstein. Subgraph isomorphism in planar graphs and related problems. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA, 1995), pages 632-640, New York, 1995. ACM.

....(extended abstract) Erik D. Demaine and MohammadTaghi Hajiaghayi Laboratory for Computer Science, Massachusetts Institute of Technology, 200 Technology Square, Cambridge, MA 02139, U.S.A. edemaine,hajiagha theory.lcs.mit.edu Abstract. We solve an open problem posed by Eppstein in 1995 [16, 17] and re enforced by Grohe [18, 19] concerning locally bounded treewidth in minor closed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a function of r (not n) Eppstein characterized ....

....times of 2 for many problems on planar graphs, such as maximum independent set, minimum dominating set, and minimum vertex cover. Chen [10] later generalized Baker s approach to obtain PTASs for K 3,3 minor free graphs and K 5 minor free graphs, but only for maximization problems. Eppstein [17, 16] further generalized Baker s approach by replacing local regions of bounded outerplanarity with local regions of bounded treewidth. The resulting approximation algorithms apply to any graph of bounded local treewidth, a notion newly introduced by Eppstein [16, 17] A graph has bounded local ....

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David Eppstein. Subgraph isomorphism in planar graphs and related problems. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA, 1995), pages 632--640, New York, 1995. ACM.

....G i , there exists a vertex v i such that each vertex in G i is at distance at most j i 1 from v i . It is known that if a planar graph has a rooted spanning tree T in which the longest path has length d, then a tree decomposition of the graph with width at most 3d can be found in time O(dn) [Bak94,Epp99]. Since each vertex in G i is at distance at most j i 1 from v i , by breadth rst search we can construct a spanning tree rooted at v i with the longest path of length at most j i 1. Hence we can construct a tree decomposition for G i of treewidth 3(j i 1) in time O( j i 1) jV (G i )j) ....

....1 1= log n) or 1 2= log n) for dominating set) As both 1= log n) and 2= log n) decrease as n increases, the solutions converge toward optimal as n increases. ut 6. 3 Further applications Our techniques are applicable to other problems solved by Eppstein using the adaptation of Baker s approach [Epp99]. For example, there exists an algorithm that determines whether a xed pattern H is a subgraph of a single crossing minor free graph 23 G in O(2 O(jV (H)j log jV (H)j) jV (G)j) time. The algorithm makes use of locally bounded treewidth; since H is of constant size, if it is a subgraph of G, ....

David Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl., 3(3):1-27, 1999.

....G i , there exists a vertex v i such that each vertex in G i is at distance at most j Gamma i 1 from v i . It is known that if a planar graph G has a rooted spanning tree T in which the longest path has length d, then a tree decomposition of G with width at most 3d can be found in time O(dn) [Bak94,Epp99]. Since each vertex in G i is at distance at most j Gamma i 1 from v i , by breadth first search, we can construct a spanning tree rooted at v i with the longest path of length at most j Gamma i 1. Hence we can construct a tree decomposition for G i of treewidth 3(j Gamma i 1) in time ....

....paper such as finding diameter if we know the graph has bounded diameter, h clustering for constant h, finding girth if we know the graph has bounded girth can be tested in O(n) for clique sum graphs. Proof. The ideas mainly follow from Baker s approach which are introduced in Eppstein s paper [Epp99]. If a graph contains a subgraph with constant size, this subgraph must be included in a constant number of consecutive layers introduced in Theorems 12 and 14. By O(n) times checking subgraph isomorphism, each consists of solving this problem for a fixed pattern H opposed a graph with bounded ....

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David Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl., 3:no. 3, 27 pp. (electronic), 1999.

....with (H) 3 and graph G have bounded treewidth with (G) 2. The subgraph isomorphism problem for the source graph G and the host graph H is NP complete. One interesting consequence of Theorem 4 is that it duplicates Eppstein s result on testing subgraph isomorphism for xed patterns [11, 12], restricted to graphs of locally bounded treewidth. Corollary 2. For a xed pattern G and a graph H of minor closed family of graphs of locally bounded treewidth, subgraph isomorphism and induced subgraph isomorphism can be tested in O(jV (H)j) time. Using this result, Eppstein also showed ....

David Eppstein. Subgraph isomorphism in planar graphs and related problems. Journal of Graph Algorithms and Applications, 3(3):27 pp. (electronic), 1999.

....has locally bounded treewidth if the treewidth of the subgraph induced on all vertices at distance r Introduction 3 from v, for any vertex v of the graph and any r # N, is bounded by a function ltw G (r) Here the function ltw G (r) the local treewidth, is dependent only on r. Eppstein [Epp99] characterized graphs of locally bounded treewidth. He also proved that Baker s results can be extended to graphs of locally bounded treewidth. In fact, a planar graph G has locally bounded treewidth with ltw G (r) 3r 1 [Bod98] Eppstein [Epp99] also extended Baker s approach to other ....

....treewidth, is dependent only on r. Eppstein [Epp99] characterized graphs of locally bounded treewidth. He also proved that Baker s results can be extended to graphs of locally bounded treewidth. In fact, a planar graph G has locally bounded treewidth with ltw G (r) 3r 1 [Bod98] Eppstein [Epp99] also extended Baker s approach to other problems on graphs of locally bounded treewidth such as the subgraph isomorphism problem for a fixed pattern G. Since, except for planar graphs, the known local treewidth for graphs of locally bounded treewidth is immense, Eppstein s polynomial time ....

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David Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl., 3:no. 3, 27 pp. (electronic), 1999.

....theorems, see [38] Of course, an improvement of the presented algorithms can also be gained by increasing the number of parameterized problems with (small) linear problem kernel. Let us nally mention that separator based techniques for solving graph problems were also used in other recent papers [16, 20]. Last but not least, our techniques might be applicable to non planar graphs, as well. This is strongly indicated by [6, 8, 21, 25] Acknowledgment We thank the anonymous referees of the 7th Annual International Computing and Combinatorics Conference (COCOON 2001) for their remarks that helped ....

D. Eppstein. Subgraph isomorphism in planar graphs and related problems. Journal of Graph Algorithms and Applications, 3(3):1-27, 1999.

....theorems, see [38] Of course, an improvement of the presented algorithms can also be gained by increasing the number of parameterized problems with (small) linear problem kernel. Let us nally mention that separator based techniques for solving graph problems were also used in other recent papers [16, 20]. Last but not least, our techniques might be applicable to non planar graphs, as well. This is strongly indicated by [6, 8, 21, 25] Acknowledgment We thank the anonymous referees of the 7th Annual International Computing and Combinatorics Conference (COCOON 2001) for their remarks that helped ....

D. Eppstein. Subgraph isomorphism in planar graphs and related problems. Journal of Graph Algorithms and Applications, 3(3):1-27, 1999.

....each g di , determine whether it is a subgraph of G. 2) Let D be the set of design concepts whose associated state transitional graphs are subgraphs of G. Find a subset of D such that it covers the entire graph G with minimum cost. Subproblem (1) us a well studied subgraph isomorphism problem (Eppstein, 1994). 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 . We denote (P j : 1 j n ) by I and write I = 1,2, m , J = 1,2, ....

Eppstein, D. (1994). "Subgraph Isomorphism in Planar Graphs and Related Problems." Technical Report 94-25, Dept. of Information and Computer Science, University of California, Irvine.

....an arbitrary large K . In general, there are various classes of graphs for which the subgraph isomorphism problem can be solved much more efficiently, without postulating upper bounds. For instance, if G and H are planar graphs the problem can be solved in time linear in the number of nodes of G (Eppstein 1999). 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. 3 Outline of the ....

Eppstein, D. (1999), Subgraph Isomorphism in Planar Graphs and Related Problems, J. Graph Algorithms and Applications 3(3):1-27.

....determining whether a graph has genus g is NP complete. As a result, for the rest of this paper, we will assume that we are given a graph and its layout of genus g. Graphs of bounded genus have several applications including VLSI layout, via minimization and bounded thickness book embeddings (see [DR91, Th89, Ep95] and the references cited therein) Informally, for a fixed ffi 0, a ffi near planar graph is a graph with vertex set V together with a planar layout with ffi Delta jV j crossovers of edges. The class of ffi near planar graphs is a generalization of planar graphs. We were motivated to study ....

....graphs are solvable efficiently when restricted to graphs of fixed genus. For example, Miller [Mi80] showed that the isomorphism problem can be solved efficiently for graphs of fixed genus. Djidjev [Dj85] gave a linear time algorithm to find small separators in bounded genus graphs. Eppstein [Ep95] gave linear time algorithms for solving the subgraph isomorphism problem for bounded genus graphs. 4 Basic definitions In this section, we present a few basic definitions which will be used in the remaining sections of the paper. 4.1 Satisfiability problems and reductions Using the general ....

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D. Eppstein, "Subgraph Isomorphism in Planar Graphs and Related Problems," 6th ACM-SIAM Symposium on Discrete Algorithms (SODA), 1995, pp. 632-640.

....of graphs admitting such a covering algorithm locally treedecomposable (a precise de nition is given in Section 4.2) Examples of locally tree decomposable classes of graphs are all classes of bounded genus, bounded valence, bounded crossing number, and, trivially, bounded tree width. Eppstein [Epp95] considered a closely related, though slightly weaker concept he called the diameter treewidth property (we call this property locally bounded tree width and refer the reader to Section 4.2 for a discussion of the various concepts) Eppstein proved that the subgraph isomorphism problem for a xed ....

.... properties is AW[1] complete) The proof of our theorem combines three main ingredients: a re nement of Courcelle s Theorem [Cou90] mentioned above, Gaifman s Theorem [Gai82] stating that rst order properties are local, and algorithmic techniques based on ideas of Baker [Bak94] and Eppstein [Epp95]. 2 Preliminaries In this paper we will con ne our attention to properties of simple undirected graphs. We consider a graph as a relational structure G = V G ; E G ) where V G is a nite set of vertices and E G is a binary relation on V G . For a subset U V G we let hUi G denote ....

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D. Eppstein. Subgraph isomorphism in planar graphs and related problems. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 632-640, 1995.

....and [2] for recent results and references. In the case of planar graphs Itai and Rodeh [16] give an O(n) algorithm for nding a triangle in the graph, if one exists (and thus solves the girth problem for planar graphs in case girth(G) 3 in O(n) time) Their results were generalized by Eppstein [11], who developed an O(n) algorithm for nding the girth of a planar graph G provided girth(G) O(1) His algorithm, however, is superexponential with respect to girth(G) This work was partially supported by the EPA grant R82 5207 01 0, EPSRC grant GR M60750, and RTDF grant 98 99 0140. Note ....

....O(n 1=2 ) ut 4 Finding shortest cycles in graphs of small girth In this section we assume that the girth of the input graph is smaller than certain parameter whose value will be determined in Section 5. For the proof of the next lemma we use a technique developed by Baker [3] and Eppstein [11]. Lemma 2. Let G be an n vertex planar graph and let d be any integer. Then we can nd in O(n) time a set of subgraphs G i of G with the following properties: i) The sum of the sizes of all subgraphs G i is O(n) ii) Every subgraph of G i has a separator of size O(d) iii) Any cycle of ....

David Eppstein. Subgraph isomorphism for planar graphs and related problems. In Proc. 6th Symp. Discrete Algorithms, pages 632-640. Assoc. Comput. Mach. and Soc. Industrial & Applied Math., 1995.

....and Greene [FG88] devised an approximation scheme for a geometric location problem related to clustering. Jiang and Wang [JW94] presented an approximation scheme for the STEINER TREE problem in the plane when the given set of regular points is c local (also called civilized) Recently, Eppstein [Ep95] obtained efficient algorithms for the SUBGRAPH ISOMORPHISM problem for graphs of fixed genus. In [MHR94, MR 97, MH 94] we investigated the existence and or non existence of polynomial time approximations and approximation schemes for several PSPACE hard problems for hierarchically specified ....

D. Eppstein, "Subgraph Isomorphism in Planar Graphs and Related Problems," Proc. 6th ACM-SIAM Symposium on Discrete Algorithms (SODA), 1995, pp. 632--640.

....g d i , determine whether it is a subgraph of G. 2) Let D 0 be the set of design concepts whose associated state transitional graphs are subgraphs of G. Find a subset of D 0 such that it covers the entire graph G with minimum cost. Subproblem (1) is a well studied subgraph isomorphism problem [13, 5]. The work by Eppstein [5] indicates that the subgraph isomorphism problem in planar graphs can be solved in linear time. To solve subproblem (2) we transform the problem into the set covering problem. Before we describe the transformation process, let us first define the set covering problem. ....

....a subgraph of G. 2) Let D 0 be the set of design concepts whose associated state transitional graphs are subgraphs of G. Find a subset of D 0 such that it covers the entire graph G with minimum cost. Subproblem (1) is a well studied subgraph isomorphism problem [13, 5] The work by Eppstein [5] indicates that the subgraph isomorphism problem in planar graphs can be solved in linear time. To solve subproblem (2) we transform the problem into the set covering problem. Before we describe the transformation process, let us first define the set covering problem. Set Covering Problem: In ....

David Eppstein, Subgraph isomorphism in planar graphs and related problems, Technical Report 94-25, Dept. of Information and Computer Science, University of California, Irvine, 1994.

....A (directed or undirected) C 5 in G, if one exists, can be found in O(E Delta (d(G) 2 ) worst case time. As a corollary, we get that if C is a non trivial minor closed family of graphs and G = V; E) is a member of C, then a C 5 in a G, if one exists, can be found in O(V ) time. Eppstein [Epp95] showed recently that if G = V; E) is a planar graph and H is a graph on k vertices, then a copy of H in G, if one exists, can be found in O(k O(k) V ) time. Eppstein s result also applies to graphs of a bounded genus but it does not apply, like our method, to all minor closed families of ....

D. Eppstein. Subgraph isomorphism in planar graphs and related problems. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, California, pages 632--640, 1995.

....method used here yields, in particular, a way of finding C k s for k 7, in O(V ) worst case time. Sundaram and Skiena [14] have recently presented some more fixed subgraph isomorphism algorithms. The results presented here, and in [1] and [15] improve some of their results. Eppstein [7] has recently shown that the fixed subgraph isomorphism problem for planar graphs, i.e. given a fixed graph H and a planar graph G = V; E) find a subgraph of G isomorphic to H , can be solved, for every fixed H , in O(V ) time. 3 Finding cycles in sparse graphs Monien [10] obtained his O(VE) ....

D. Eppstein. Subgraph isomorphism in planar graphs and related problems. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, California, pages 632--640, 1995.

....naturally raises the question, for which other minor closed graph families can we prove a bound relating diameter to tree width, similar to Lemma 4 Such a result would then let us apply our subgraph isomorphism techniques unchanged to any such families. In the conference version of this paper [18], we announced an exact characterization of these families, which are detailed in a separate journal paper [15] and which we now summarize: De nition 3 De ne a family F of graphs to have the diameter treewidth property if there is some function f(D) such that every graph in F with diameter at ....

D. Eppstein. Subgraph isomorphism in planar graphs and related problems. Proc. 6th Symp. Discrete Algorithms, pp. 632-640. Assoc. Comput. Mach. and Soc. Industrial & Applied Math., 1995.

....certificate construction method from our companion paper, together with the certificates defined here, gives a unified method of testing 3 and 4 edge , and 2 and SPARSIFICATION II: EDGE AND VERTEX CONNECTIVITY 343 3 vertex connectivity in planar graphs, in linear time. In recent work, Eppstein [7] has shown how to compute k edge or k vertex connectivity in planar graphs in linear time for any constant k. The remainder of this paper consists of the following sections. Section 2 contains basic definitions. In section 3 we recall some properties of separator based sparsification and ....

D. Eppstein, Subgraph isomorphism for planar graphs and related problems, in Proc. 6th ACMSIAM Symp. on Discrete Algorithms, 1995, pp. 189--196; J. Graph Algorithms Appl., to appear.

....naturally raises the question, for which other minor closed graph families can we prove a bound relating diameter to tree width, similar to Lemma 4 Such a result would then let us apply our subgraph isomorphism techniques unchanged to any such families. In the conference version of this paper [18], we announced an exact characterization of these families, which are detailed in a separate journal paper [15] and which we now summarize: De nition 3 De ne a family F of graphs to have the diameter treewidth property if there is some function f(D) such that every graph in F with diameter at ....

D. Eppstein. Subgraph isomorphism in planar graphs and related problems. Proc. 6th Symp. Discrete Algorithms, pp. 632-640. Assoc. Comput. Mach. and Soc. Industrial & Applied Math., 1995.

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D. Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms & Applications, 3(3):1--27,1999.

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David Eppstein. Subgraph isomorphism in planar graphs and related problems. Journal of Graph Algorithms and Applications, 3(3):27 pp. (electronic), 1999.

No context found.

David Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl., 3(3):1-27, 1999.

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D. Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms & Applications, 3(3):1--27,1999.

No context found.

D. Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms & Applications, 3(3):1--27,1999.

No context found.

D. Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms & Applications, 3(3):1--27,1999.

No context found.

David Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl., 3(3):1--27, 1999.

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D. EPPSTEIN, Subgraph isomorphism in planar graphs and related problems, in Proceedings of the Sixth Annual ACMSIAM Symposium on Discrete Algorithms (San Francisco, CA,

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D. Eppstein. Subgraph isomorphism in planar graphs and related problems. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms and Applications, volume 3, pages 1-27, 1995.

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D. Eppstein, "Subgraph isomorphism in planar graphs and related problems," tech. rep., Dept. of Information and Computer Science, University of California, May 1994.

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