Courcelle, B., and M etivier, Y. Coverings and minors: Applications to local computations in graphs. European Journal of Combinatorics 15 (1994), 127-138.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....k coverings were introduced in [12] as a tool for proving impossibility results for local graph computations. There are various types of local graph computations, their relative computational power and ability (or inability) to recognize different graph families were examined in a series of papers [14, 11, 9, 3, 15, 4]. All types of local graph computations share two basic properties: 1. they are graph relabelling systems, i.e. they do not transform the structure of the underlying graph but only modify the labels of vertices edges, 2. the label of any vertex v is modified only locally according to a fixed ....

....that a given family of connected graphs F is not recognizable by local computations it suffices to show that there is a graph G in F with a k covering not belonging to F . However, to apply this method in practice we need a technique of constructing describing k coverings (see final remarks in [4]) It is interesting to note that the balls of radius k appear also in the characterization of first order definable families of graphs, or more generally in the definition of locally threshold testable sets of graphs, see [22] However, local graph computations are less powerful since they ....

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B. Courcelle and Y. M'etivier. Covering and minors: Application to local computations in graphs. Europ. J. of Combinatorics, 15:127--138, 1994.

.... special cases of covering spaces from algebraic topology [15] and have many applications in topological graph theory [6] In an algorithmic framework, graph coverings have been applied by Angluin to study local knowledge in distributed computing environments [2] and by Courcelle and M etivier [4] to show that nontrivial minor closed classes of graphs cannot be recognized by local computations. Abello et al. 1] raised the question of computational complexity of H cover problems, noting that there are both polynomial time solvable (easy) and NP complete (difficult) versions of this ....

B. Courcelle and Y. M'etivier, Coverings and minors: Applications to local computations in graphs, European Journal of Combinatorics 15 (1994), 127-138.

....uses covering graphs constructed from the homology group (cycle space) of G to classify cubic graphs with given symmetry types. In a more applied setting, graph coverings have been used by Angluin [2] to study local knowledge in distributed computing environments, and by Courcelle and M etivier [6] to show that nontrivial minor closed classes of graphs cannot be recognized by local computations. In [1] Abello et al. raised the question of computational complexity of H cover problems, noting that there are both polynomial time solvable (easy) and NP complete (difficult) versions of this ....

B. Courcelle and Y. M'etivier, Coverings and minors: Applications to local computations in graphs, European Journal of Combinatorics 15 (1994), 127-138

....its elements. By a minor closed class of connected graphs we mean a class of connected graphs containing all connected minors of its elements. Minor closed classes of connected graphs can be characterized by nite sets of connected forbidden minors, also called obstructions [26, 27] Then we have [28]: Theorem 62 No minor closed class of connected graphs, which is not the class of all connected graphs and contains at least one graph with at least two cycles, can be recognized by locally generated relabelling relations. Before proving this result, we need two technical lemmas. Lemma 63 For ....

B. Courcelle and Y. Metivier. Coverings and minors : application to local computations in graphs. Europ. J. Combinatorics, 15:127-138, 1994. 36 Graph relabelling systems and distributed algorithms

.... to BH (x; k) where stands for the k covering of (H; by (G; Since there exist planar graphs with non planar k coverings for every k, we get that the class of planar graphs cannot be recognized, even in an undeterministic way [13] Using this notion of k covering, it is proved in [5] that every non trivial minor closed class of graphs containing at least one graph with at least two cycles cannot be recognized by a k locally generated graph relabelling relation. A standard method for producing coverings of a graph G is to consider the kronecker product of G by the complete ....

Courcelle, B.|Metivier, Y.: Coverings and minors : application to local computations in graphs. Europ. J. Combin. 15:127-138, 1994.

....x. Let k be a positive integer. We say that a graph G is a k covering of a graph G 0 via a mapping from V (G) onto V (G 0 ) if is a surjective homomorphism such that for every vertex v of V (G) the restriction of to BG (v; k) is an isomorphism between BG (v; k) and BG 0 ( v) k) In [4] the following is proved : Theorem 6 [4] Every class of connected graphs recognizable by local computations is closed under coverings. Using that, we easily obtain : Theorem 7 Let A and B be two labels, let m 0 be an integer ; the class of labelled connected graphs G such that jGj A jGj B m ....

.... a graph G is a k covering of a graph G 0 via a mapping from V (G) onto V (G 0 ) if is a surjective homomorphism such that for every vertex v of V (G) the restriction of to BG (v; k) is an isomorphism between BG (v; k) and BG 0 ( v) k) In [4] the following is proved : Theorem 6 [4] Every class of connected graphs recognizable by local computations is closed under coverings. Using that, we easily obtain : Theorem 7 Let A and B be two labels, let m 0 be an integer ; the class of labelled connected graphs G such that jGj A jGj B m (resp. jGj A = jGj B m) is not ....

B. Courcelle and Y. Metivier, Coverings and minors : application to local computations in graphs, Europ. J. Combinatorics 15 (1994), 127-138.

....Malostransk e n am. 25, 118 00 Prague, Czech Republic, e mail: honza kam.ms.mff.cuni.cz. Research supported in part by Czech Research grants KONTAKT 338 99, KONTAKT 055 97 and GAUK 158 99. 1 [2] to study local knowledge in distributed computing environments, and by Courcelle and M etivier [6] to show that nontrivial minor closed classes of graphs cannot be recognized by local computations. In [1] Abello et al. raised the question of computational complexity of H cover problems, noting that there are both polynomial time solvable (easy) and NPcomplete (dicult) versions of this ....

B. Courcelle and Y. Metivier, Coverings and minors: Applications to local computations in graphs, European Journal of Combinatorics 15 (1994), 127-138

....as graph relabelling systems with priorities (PGRS) The computational power of Noetherian graph relabelling systems with priorities is studied in [LM93] In particular, the classes of sets of graphs recognized by PGRS are compared with the classes definable by logic formulas. It is proved in [CM94] that, apart from a few exceptions, minor closed classes of graphs cannot be recognized by graph relabelling systems. Planar graphs and graphs of tree width at most k are typical examples of minor closed classes of graphs. In [MS94] the well known problem of election in a tree is considered, and ....

B. Courcelle and Y. Metivier. Coverings and minors: Application to local computations in graphs. European J. of Combinatorics, 15:127--138, 1994.

....The implementation of PGRS is described in [Bil91] The computational power of noetherian graph relabelling systems with priorities is studied in [LM93] In particular, the classes of sets of graphs recognized by PGRS are compared with the classes definable by logic formulas. It is proved in [CM94] that, apart from a few exceptions, minor closed classes of graphs cannot be recognized by graph relabelling systems. Planar graphs and graphs of bounded tree width are typical examples of minor closed classes of graphs. In [MS94] the well known problem of election in a tree is considered, and ....

B. Courcelle and Y. Metivier. Coverings and minors: application to local computations in graphs. European J. of Combinatorics, 15:127--138, 1994.

....we present a formalization of the local detection problem and we introduce some new methods for obtaining impossibility results. More precisely, we extend the notion of coverings, which is known from algebraic topology [13] and has been already used in distributed computing for negative results [1,5,7,9], to quasi coverings. Quasi coverings capture some topologies which fail to cope with the classical coverings. We show in this paper that one cannot detect locally the global termination for uniformly labelled graphs belonging to certain families of connected graphs C. More specifically, it ....

B. Courcelle and Y. M'etivier, Coverings and minors : application to local computations in graphs, Europ. J. Combinatorics 15 (1994) 127--138.

....k coverings were introduced in [12] as a tool for proving impossibility results for local graph computations. There are various types of local graph computations, their relative computational power and ability (or inability) to recognize different graph families were examined in a series of papers [14, 11, 9, 3, 15, 4]. All types of local graph computations share two basic properties: 1. they are graph relabelling systems, i.e. they do not transform the structure of the underlying graph but only modify the labels of vertices edges, 2. the label of any vertex v is modified only locally according to a fixed set ....

....that a given family of connected graphs F is not recognizable by local computations it suffices to show that there is a graph G in F with a k covering not belonging to F . However, to apply this method in practice we need a technique of constructing describing k coverings (see final remarks in [4]) It is interesting to note that the balls of radius k appear also in the characterization of first order definable families of graphs, or more generally in the definition of locally threshold testable sets of graphs, see [22] However, local graph computations are less powerful since they ....

[Article contains additional citation context not shown here]

B. Courcelle and Y. M'etivier. Covering and minors: Application to local computations in graphs. Europ. J. of Combinatorics, 15:127--138, 1994.

....locally bijective. A graph is ambiguous, if there exists a labelling of its nodes which is locally bijective, but not bijective. Ambiguity of graphs is another formulation of the existence of a non trivial covering for a graph, the property introduced and investigated by Courcelle and M etivier in [1]. It has been proved that ambiguity of graphs is the source of some impossiblility results given in [4] In consequence, enumeration protocol cannot exist for ambiguous graphs; until now, however, the converse has not been proved, namely, no local enumeration protocol for the whole class of ....

B. Courcelle and Y. M'etivier, Coverings and minors: application to local computations on graphs, Europ. J. of Combinatorics 15 (1994) 127-138.

....a positive integer. We say that a graph G is a k Gammacovering of a graph G 0 via a mapping fl from V (G) onto V (G 0 ) if fl is a surjective homomorphism such that for every vertex v of V (G) the restriction of fl to BG (v; k) is an isomorphism between BG (v; k) and BG 0 (fl(v) k) In [4] the following is proved : Theorem 4.1 [4] Every class of connected graphs recognizable by local computations is closed under coverings. Using that, we easily obtain : Theorem 4.2 Let A and B be two labels, let m 0 be an integer ; the class of labelled connected graphs G such that jGj A jGj ....

....is a k Gammacovering of a graph G 0 via a mapping fl from V (G) onto V (G 0 ) if fl is a surjective homomorphism such that for every vertex v of V (G) the restriction of fl to BG (v; k) is an isomorphism between BG (v; k) and BG 0 (fl(v) k) In [4] the following is proved : Theorem 4. 1 [4] Every class of connected graphs recognizable by local computations is closed under coverings. Using that, we easily obtain : Theorem 4.2 Let A and B be two labels, let m 0 be an integer ; the class of labelled connected graphs G such that jGj A jGj B Gamma m (resp. jGj A = jGj B Gamma m) ....

B. Courcelle and Y. M'etivier, Coverings and minors : application to local computations in graphs, Europ. J. Combinatorics 15 (1994), 127--138.

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Courcelle, B., and M etivier, Y. Coverings and minors: Applications to local computations in graphs. European Journal of Combinatorics 15 (1994), 127-138.

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Bruno Courcelle and Yves Metivier. Coverings and minors: Applications to local computations in graphs. European Journal of Combinatorics, 15:127-138, 1994.

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Bruno Courcelle and Yves Metivier. Coverings and minors: Applications to local computations in graphs. European Journal of Combinatorics, 15:127--138, 1994.

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B. Courcelle and Y. Mtivier. Coverings and minors : application to local computations in graphs. Europ. J. Combinatorics, 15:127138, 1994.

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B. Courcelle, Y. M'etivier, Coverings and minors: application to local computations in graphs, European Journal of Combinatorics 15 (1994), 127--138

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B. Courcelle and Y. M'etivier, Coverings and minors: application to local computations on graphs, Europ. J. of Combinatorics 15 (1994) 127-138.

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