N. Alon, On the Edge-Expansion of Graphs, Combinatorics, Probability and Computing, vol.6, issue.2, pp.1-10, 1993.
DOI : 10.1017/S096354839700299X

N. Alon and M. Capalbo, Smaller Explicit Superconcentrators, Proc 14th Ann Symp Discret Algorithms ACM-SIAM SODA, pp.340-346, 2003.
DOI : 10.1080/15427951.2004.10129083

N. Alon, Z. Galil, and V. D. Milman, Better expanders and superconcentrators, Journal of Algorithms, vol.8, issue.3, pp.337-347, 1987.
DOI : 10.1016/0196-6774(87)90014-9

N. Alon, P. Hamburger, and A. V. Kostochka, Regular Honest Graphs, Isoperimetric Numbers, and Bisection of Weighted Graphs, European Journal of Combinatorics, vol.20, issue.6, pp.469-481, 1999.
DOI : 10.1006/eujc.1998.0295

N. Alon and V. D. Milman, Eigenvalues, Expanders And Superconcentrators, 25th Annual Symposium onFoundations of Computer Science, 1984., pp.320-322, 1984.
DOI : 10.1109/SFCS.1984.715931

L. A. Bassalygo, Asymptotically optimal switching circuits, Problemy Pederachi Informatsii, vol.17, pp.81-88, 1981.

B. Beauquier and E. Darrot, Arbitrary size Waksman networks,Premì ere rencontres francophones sur les aspects algorithmiques de télécommunication (Algotel), pp.95-100, 1999.

B. Beauquier and E. Darrot, Arbitrary size Waksman networks and their vulnerability , Parallel Process Lett 3, pp.287-296, 2002.

J. C. Bermond, E. Darrot, and O. Delmas, Design of fault-tolerant networks for satellites (TWTA redundancy), Networks, vol.40, issue.4, pp.202-207, 2002.
DOI : 10.1002/net.10044
URL : https://hal.archives-ouvertes.fr/hal-00307611

J. C. Bermond, O. Delmas, F. Havet, M. Montassier, and S. Perennes, Réseaux de télécommunications minimaux embarqués tolérants aux pannes,Cinquì eme rencontres francophones sur les aspects algorithmiques de télécommunication, pp.27-32, 2003.

J. C. Bermond, F. Giroire, and S. Pérennes, Design of Minimal Fault Tolerant On-Board Networks: Practical Constructions, 14th Int Colloq Structural Informat Commun Complexity (SIROCCO), pp.261-273, 2007.
DOI : 10.1007/978-3-540-72951-8_21
URL : https://hal.archives-ouvertes.fr/hal-00512282

J. C. Bermond, F. Havet, and C. D. Tóth, Fault tolerant on-board networks with priorities, Networks, vol.6, issue.1, pp.47-56, 2006.
DOI : 10.1002/net.20094
URL : https://hal.archives-ouvertes.fr/inria-00070640

Y. Bilu and N. Linial, Lifts, Discrepancy and Nearly Optimal Spectral Gap*, Combinatorica, vol.26, issue.5, pp.495-519, 2006.
DOI : 10.1007/s00493-006-0029-7

M. Blum, R. Karp, C. Papadimitriou, O. Vornberger, and M. Yannakakis, The complexity of testing whether a graph is a superconcentrator, Information Processing Letters, vol.13, issue.4-5, pp.164-167, 1981.
DOI : 10.1016/0020-0190(81)90050-8

B. Bollobás, The Isoperimetric Number of Random Regular Graphs, European Journal of Combinatorics, vol.9, issue.3, pp.241-244, 1988.
DOI : 10.1016/S0195-6698(88)80014-3

M. Capalbo, O. Reingold, S. Vadhan, and A. Wigderson, Randomness conductors and constant-degree lossless expanders, 34th ACM Symp Theory Comput (STOCS), pp.659-668, 2002.

F. R. Chung, On Concentrators, Superconcentrators, Generalizers, and Nonblocking Networks, Bell System Technical Journal, vol.58, issue.8, pp.1765-1777, 1979.
DOI : 10.1002/j.1538-7305.1979.tb02972.x

F. R. Chung, Spectral Graph Theory, 1997.
DOI : 10.1090/cbms/092

G. Davidoff, P. Sarnak, and A. Valette, Elementary number theory, group theory, and Ramanujan graphs, 2003.
DOI : 10.1017/CBO9780511615825

O. Delmas, F. Havet, M. Montassier, and S. Pérennes, Design of fault tolerant onboard networks, INRIA Res Report, vol.5866, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01111370

F. Giroire, Réseaux, algorithmique et analyse combinatoire de grands ensembles, 2006.

C. Greenhill, J. H. Kim, and N. C. Wormald, Hamiltonian decompositions of random bipartite regular graphs, Journal of Combinatorial Theory, Series B, vol.90, issue.2, pp.195-222, 2004.
DOI : 10.1016/j.jctb.2003.07.001

F. Havet, REPARTITORS, SELECTORS AND SUPERSELECTORS, Journal of Interconnection Networks, vol.07, issue.03, pp.391-415, 2006.
DOI : 10.1142/S0219265906001752
URL : https://hal.archives-ouvertes.fr/inria-00070327

B. Monien and R. Preis, Upper bounds on the bisection width of 3- and 4-regular graphs, Journal of Discrete Algorithms, vol.4, issue.3, pp.475-498, 2006.
DOI : 10.1016/j.jda.2005.12.009

M. Morgenstern, Existence and Explicit Constructions of q + 1 Regular Ramanujan Graphs for Every Prime Power q, Journal of Combinatorial Theory, Series B, vol.62, issue.1, pp.44-62, 1994.
DOI : 10.1006/jctb.1994.1054

M. Murty, Ramanujan graphs, J Ramanujan Math Soc, vol.18, pp.33-52, 2003.

H. Q. Ngo and D. Du, Notes on the complexity of switching networks Advances in Switching Networks, pp.307-367, 2001.

O. Reingold, S. Vadhan, and A. Wigderson, Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors, Proceedings 41st Annual Symposium on Foundations of Computer Science, pp.157-187, 2002.
DOI : 10.1109/SFCS.2000.892006

U. Schöning, Smaller superconcentrators of density 28, Information Processing Letters, vol.98, issue.4, pp.127-129, 2006.
DOI : 10.1016/j.ipl.2006.01.006

L. G. Valiant, On non-linear lower bounds in computational complexity, Proceedings of seventh annual ACM symposium on Theory of computing , STOC '75, pp.45-53, 1975.
DOI : 10.1145/800116.803752

]. S. Alouf, E. Altman, J. Galtier, J. Lalande, and C. Touati, Quasi-optimal bandwidth allocation for multi-spot MFTDMA satellites, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies., 2005.
DOI : 10.1109/INFCOM.2005.1497923
URL : https://hal.archives-ouvertes.fr/inria-00451816

F. Havet, X. Akyildiz, W. Wang, and . Wang, Channel assignment and multicolouring of the induced subgraphs of the triangular lattice, Discrete Mathematics, vol.233, issue.1-3, pp.219-231, 2001.
DOI : 10.1016/S0012-365X(00)00241-7

H. Liu and B. Zhao, Optimal Scheduling for Link Assignment in Traffic-Sensitive STDMA Wireless Ad-Hoc Networks, ICCNMC, ser. LNCS, pp.218-228, 2005.
DOI : 10.1007/11534310_25

R. Klasing, N. Morales, and S. Perennes, On the complexity of bandwidth allocation in radio networks with steady traffic demands, Theoretical Computer Science, 2008.
URL : https://hal.archives-ouvertes.fr/inria-00070575

J. Bibb-cain, T. Billhartz, and L. , Foore A link scheduling and ad hoc networking approach using directional antennas, Military comm. conf, pp.643-648, 2003.

R. C. Carlson and G. L. Nemhauser, Scheduling to Minimize Interaction Cost, Operations Research, vol.14, issue.1, pp.52-58, 1966.
DOI : 10.1287/opre.14.1.52

I. Holyer, The NP-Completeness of Edge-Coloring, SIAM Journal on Computing, vol.10, issue.4, pp.718-720, 1981.
DOI : 10.1137/0210055

D. P. Koreas, The NP-completeness of chromatic index in triangle free graphs with maximum vertex of degree 3, Applied Mathematics and Computation, vol.83, issue.1, pp.13-17, 1997.
DOI : 10.1016/S0096-3003(96)00021-5

C. Mcdiarmid, Discrete Mathematics and Radio Channel Assignment, Linhares-Sales, pp.27-63, 2003.
DOI : 10.1007/0-387-22444-0_2

M. Padberg and M. Rao, -Matchings, Mathematics of Operations Research, vol.7, issue.1, pp.67-80, 1982.
DOI : 10.1287/moor.7.1.67
URL : https://hal.archives-ouvertes.fr/hal-00761068

V. G. Vizing, On an estimate of a chromatic class of a p-graph (In Russian), Diskret, Analiz, vol.3, pp.25-30, 1964.

N. Alon, C. Mcdiarmid, and B. Reed, Star arboricity, Combinatorica, vol.39, issue.4, pp.375-380, 1992.
DOI : 10.1007/BF01305230

R. Brandt, Multicasting using WDM in Multifiber Optical Star Networks, 2003.

R. Brandt and T. F. Gonzalez, WAVELENGTH ASSIGNMENT IN MULTIFIBER OPTICAL STAR NETWORKS UNDER THE MULTICASTING COMMUNICATION MODE, Journal of Interconnection Networks, vol.06, issue.04, pp.383-405, 2005.
DOI : 10.1142/S0219265905001484

A. Frank, Covering branchings, Acta Sci. Math. (Szeged), vol.41, pp.77-81, 1979.

B. Guiduli, On incidence coloring and star arboricity of graphs, Discrete Mathematics, vol.163, issue.1-3, pp.275-278, 1997.
DOI : 10.1016/0012-365X(95)00342-T

A. Lardies, R. Gupta, and R. Patterson, Traffic grooming in a multi-layer network, Opt. Netw. Mag, vol.2, pp.91-99, 2001.

E. Modiano and P. J. Lin, Traffic grooming in WDM networks, IEEE Communications Magazine, vol.39, issue.7, pp.124-129, 2001.
DOI : 10.1109/35.933446

A. Pinlou and E. Sopena, The acircuitic directed star arboricity of subcubic graphs is at most four, Discrete Mathematics, vol.306, issue.24
DOI : 10.1016/j.disc.2006.06.007
URL : https://hal.archives-ouvertes.fr/hal-00307095

A. Schrijver, Combinatorial optimization. Polyhedra and efficiency, Algorithms and Combinatorics, vol.24, 2003.

M. Maier, M. Herzog, and M. Reisslein, STARGATE, IEEE COMMUNICATIONS MAGAZINE, vol.45, issue.5, p.50, 2007.
DOI : 10.1017/CBO9780511619731.020

C. Barnhart, E. Johnson, G. Nemhauser, M. Savelsbergh, and P. Vance, Branch-and-Price: Column Generation for Solving Huge Integer Programs, Operations Research, vol.46, issue.3, pp.316-329, 1998.
DOI : 10.1287/opre.46.3.316

L. Shen, X. Yang, and B. Ramamurthy, Shared-risk link group (SRLG)-diverse path provisioning under hybrid service level agreements in wavelength-routed optical mesh networks: formulation and solution approaches, OptiComm 2003: Optical Networking and Communications, pp.918-931, 2005.
DOI : 10.1117/12.533306

S. Yuan, S. Varma, and J. Jue, Minimum-color path problems for reliability in mesh networks, IEEE INFOCOM, pp.2658-2669, 2005.

C. Liu and L. Ruan, p-Cycle Design in Survivable WDM Networks with Shared Risk Link Groups (SRLGs), Photonic Network Communications, vol.11, issue.3, pp.301-311, 2006.
DOI : 10.1007/s11107-005-7357-1

S. Ramamurthy and B. Mukherjee, Survivable WDM mesh networks. Part I-Protection, IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320), pp.744-751, 1999.
DOI : 10.1109/INFCOM.1999.751461
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.121.3976

D. Papadimitriou, F. Poppe, J. Jones, S. Venkatachalam, S. Dharanikota et al., Inference of shared risk link groups, IETF Draft, pp.2001-066

D. Coudert, P. Datta, S. Perennes, H. Rivano, and M. Voge, SHARED RISK RESOURCE GROUP COMPLEXITY AND APPROXIMABILITY ISSUES, Parallel Processing Letters, vol.17, issue.02, pp.169-184, 2007.
DOI : 10.1142/S0129626407002958
URL : https://hal.archives-ouvertes.fr/inria-00070167

D. Coudert, S. Perennes, H. Rivano, and M. Voge, Shared risk resource groups and colored graph: Polynomial cases and transformation issues, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00175143

M. Voge, How to Transform a Multilayer Network into a Colored Graph, 2006 International Conference on Transparent Optical Networks, pp.116-119, 2006.
DOI : 10.1109/ICTON.2006.248415

A. Faragó, A graph theoretic model for complex network failure scenarios, Proceedings of the Eighth INFORMS Telecommunications Conference, 2006.

P. Datta and A. Somani, Diverse routing for shared risk resource groups (SRRG) failures in WDM optical networks, First International Conference on Broadband Networks, pp.120-129, 2004.
DOI : 10.1109/BROADNETS.2004.34

D. Coudert, S. Perennes, H. Rivano, and M. Voge, Shared Risk Resource Groups and Survivability in Multilayer Networks, 2006 International Conference on Transparent Optical Networks, pp.235-238, 2006.
DOI : 10.1109/ICTON.2006.248442
URL : https://hal.archives-ouvertes.fr/inria-00429170

G. Even, G. Kortsarz, and W. Slany, On network design problems: fixed cost flows and the covering steiner problem, ACM Transactions on Algorithms, vol.1, issue.1, pp.74-101, 2005.
DOI : 10.1145/1077464.1077470

R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network flows: theory, algorithms, and applications, 1993.

J. Lalande, M. Syska, and Y. Verhoeven, Mascopt -a network optimization library: Graph manipulation INRIA Sophia Antipolis, Tech. Rep. RT-0293 Available: http://www-sop.inria. fr/mascotte/mascopt Routing and wavelength assignment by partition coloring, European Journal of Operational Research, vol.1713, issue.3, pp.797-810, 2004.

S. Orlowski and M. Pioro, Primal and dual formulations of network reconfiguration problems and related path generation issues, 2007.

R. Ramaswami and K. Sivarajan, Optical Networks -A Practical Perspective, 2002.

R. Dutta and G. Rouskas, A survey of virtual topology design algorithms for wavelength routed optical networks, Optical Networks Magazine, pp.73-89, 2000.

H. Zang, J. P. Jue, and B. Mukherjee, A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks, Optical Networks Magazine, pp.47-60, 2000.

B. Jaumard, C. Meyer, and B. Thiongane, Decomposition methods for the RWA problem

R. Dutta and G. Rouskas, Traffic grooming in WDM networks: past and future, IEEE Network, vol.16, issue.6, pp.46-56, 2002.
DOI : 10.1109/MNET.2002.1081765

E. Modiano and P. Lin, Traffic grooming in WDM networks, IEEE Communications Magazine, vol.39, issue.7, pp.124-129, 2001.
DOI : 10.1109/35.933446

A. Somani, Survivability and Traffic Grooming in WDM Optical Networks, 2006.
DOI : 10.1017/CBO9780511616105

K. Zhu and B. Mukherjee, A review of traffic grooming in WDM optical networks: Architectures and challenges, Optical Networks Magazine, vol.4, issue.2, pp.55-64, 2003.

J. Hu and B. Leida, Traffic grooming, routing, and wavelength assignment in optical WDM mesh networks, IEEE INFOCOM 2004, 2004.
DOI : 10.1109/INFCOM.2004.1354521

S. Huang, R. Dutta, and G. N. Rouskas, Traffic grooming in path, star, and tree networks: Complexity, bounds, and algorithms A framework for service-guaranteed shared protection in wdm mesh networks, IEEE Communications Magazine, pp.97-103, 2002.

A. Bouffard, Dimensionnement GRWA et protection par segment dans les r ? A c seaux optiques WDM, Canada, 2005.

T. Hemazro, B. Jaumard, and O. Marcotte, A column generation and branch-and-cut algorithm for the channel assignment problem, Computers and Operations Research, 2007.
DOI : 10.1016/j.cor.2006.07.012

B. Jaumard, B. Vignac, and F. Vanderbeck, An efficient global search heuristic for the grwa proble, 2007.

D. Coudert and J. Sereni, Characterization of graphs and digraphs with small process numbers, Discrete Applied Mathematics, vol.159, issue.11, 2007.
DOI : 10.1016/j.dam.2011.03.010

J. Díaz, J. Petit, and M. Serna, A survey of graph layout problems, ACM Computing Surveys, vol.34, issue.3, pp.313-356, 2002.
DOI : 10.1145/568522.568523

J. A. Ellis, I. H. Sudborough, and J. S. Turner, The Vertex Separation and Search Number of a Graph, Information and Computation, vol.113, issue.1, pp.50-79, 1994.
DOI : 10.1006/inco.1994.1064

F. V. Fomin and D. Thilikos, An annotated bibliography on guaranteed graph searching, Theoretical Computer Science, vol.399, issue.3, 2008.
DOI : 10.1016/j.tcs.2008.02.040

N. G. Kinnersley, The vertex separation number of a graph equals its path-width, Information Processing Letters, vol.42, issue.6, pp.345-350, 1992.
DOI : 10.1016/0020-0190(92)90234-M

M. Kirousis and C. H. Papadimitriou, Searching and pebbling, Theoretical Computer Science, vol.47, issue.2, pp.205-218, 1986.
DOI : 10.1016/0304-3975(86)90146-5

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou, The complexity of searching a graph, Journal of the ACM, vol.35, issue.1, pp.18-44, 1988.
DOI : 10.1145/42267.42268

T. D. Parsons, Pursuit-evasion in a graph, Theory and applications of graphs, pp.426-441, 1978.
DOI : 10.1007/BFb0070400

N. Robertson and P. D. Seymour, Graph minors. I. Excluding a forest, Journal of Combinatorial Theory, Series B, vol.35, issue.1, pp.39-61, 1983.
DOI : 10.1016/0095-8956(83)90079-5
URL : http://doi.org/10.1006/jctb.1999.1919

P. Scheffler, A Linear Algorithm for the Pathwidth of Trees, Topics in Combinatorics and Graph Theory, pp.613-620, 1990.
DOI : 10.1007/978-3-642-46908-4_70

K. Skodinis, Construction of linear tree-layouts which are optimal with respect to vertex separation in linear time, Journal of Algorithms, vol.47, issue.1, pp.40-59, 2003.
DOI : 10.1016/S0196-6774(02)00225-0

. Unité-de-recherche-inria-sophia and . Antipolis, route des Lucioles -BP 93 -06902 Sophia Antipolis Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université -ZAC des Vignes 4, 2004.

I. Unité-de-recherche and . Lorraine, Technopôle de Nancy-Brabois -Campus scientifique 615, rue du Jardin Botanique -BP 101 -54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu -35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l'Europe -38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt, Domaine de Voluceau -Rocquencourt -BP 105 -78153 Le Chesnay Cedex

I. De-voluceau-rocquencourt, BP 105 -78153 Le Chesnay Cedex (France) http://www.inria.fr ISSN, pp.249-6399

S. Antipolis and F. E. , -mail: lastname@sophia.inria.fr Figure 3: G 2 , four disjoint copies of G 1 glued with a grey cross K, and its weak dual

]. O. Amini, F. Huc, and S. Pérennes, On the pathwidth of planar graphs, 2006.
URL : https://hal.archives-ouvertes.fr/inria-00082035

H. L. Bodlaender, A partial k-arboretum of graphs with bounded treewidth, Theoretical Computer Science, vol.209, issue.1-2, pp.1-45, 1998.
DOI : 10.1016/S0304-3975(97)00228-4

H. L. Bodlaender and F. V. Fomin, Approximation of pathwidth of outerplanar graphs, Journal of Algorithms, vol.43, issue.2, pp.190-200, 2002.
DOI : 10.1016/S0196-6774(02)00001-9

H. L. Bodlaender and T. Kloks, Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs, Journal of Algorithms, vol.21, issue.2, pp.358-402, 1996.
DOI : 10.1006/jagm.1996.0049

V. Bouchitté, F. Mazoit, and I. Todinca, Chordal embeddings of planar graphs, Discrete Mathematics, vol.273, issue.1-3, pp.85-10201, 2003.
DOI : 10.1016/S0012-365X(03)00230-9

J. Díaz, J. Petit, and M. Serna, A survey of graph layout problems, ACM Computing Surveys, vol.34, issue.3, pp.313-356, 2002.
DOI : 10.1145/568522.568523

J. Ellis and M. Markov, Computing the vertex separation of unicyclic graphs, Information and Computation, vol.192, issue.2, pp.123-161, 2004.
DOI : 10.1016/j.ic.2004.03.005

J. A. Ellis, I. H. Sudborough, and J. S. Turner, The Vertex Separation and Search Number of a Graph, Information and Computation, vol.113, issue.1, pp.50-79, 1994.
DOI : 10.1006/inco.1994.1064

F. Fomin and D. M. Thilikos, On self duality of pathwidth in polyhedral graph embeddings, Journal of Graph Theory, vol.14, issue.1, 2006.
DOI : 10.1002/jgt.20219

F. V. Fomin, Pathwidth of Planar and Line Graphs, Graphs and Combinatorics, vol.19, issue.1, pp.91-99, 2003.
DOI : 10.1007/s00373-002-0490-z

R. Govindan, M. A. Langston, and X. Yan, Approximating the pathwidth of outerplanar graphs, Information Processing Letters, vol.68, issue.1, pp.17-23, 1998.
DOI : 10.1016/S0020-0190(98)00139-2

N. G. Kinnersley, The vertex separation number of a graph equals its path-width, Information Processing Letters, vol.42, issue.6, pp.345-350, 1992.
DOI : 10.1016/0020-0190(92)90234-M

D. Lapoire, Structuration des graphes planaires, 1999.

N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou, The complexity of searching a graph, Journal of the ACM, vol.35, issue.1, pp.18-44, 1988.
DOI : 10.1145/42267.42268

S. L. Mitchell, Linear algorithms to recognize outerplanar and maximal outerplanar graphs, Information Processing Letters, vol.9, issue.5, pp.229-232, 1979.
DOI : 10.1016/0020-0190(79)90075-9

B. Reed, Treewidth and tangles: an new connectivity measure and some applications, Surveys in Combinatorics, pp.87-162, 1997.

N. Robertson and P. D. Seymour, Graph minors. III. Planar tree-width, Journal of Combinatorial Theory, Series B, vol.36, issue.1, pp.49-64, 1984.
DOI : 10.1016/0095-8956(84)90013-3
URL : http://doi.org/10.1006/jctb.1999.1919

P. Scheffler, A Linear Algorithm for the Pathwidth of Trees, Topics in Combinatorics and Graph Theory, pp.613-620, 1990.
DOI : 10.1007/978-3-642-46908-4_70

M. M. Sys?o, Characterizations of outerplanar graphs, Discrete Mathematics, vol.26, issue.1, pp.47-53, 1979.
DOI : 10.1016/0012-365X(79)90060-8

M. Yannakakis, A polynomial algorithm for the min-cut linear arrangement of trees, Journal of the ACM, vol.32, issue.4, pp.950-988, 1985.
DOI : 10.1145/4221.4228

G. Let, H. Be, and . Graphs, If there exists a connected map ? of degree at most k from G to H, we have: pw(G) ? k · pw

@. For-every-edge-xy and ?. , there is one ? ?1 (X i ) which contains both x and y. To prove this, let A = ?(x) and B = ?(y) The graph induced by H on A ? B is connected, so at least one of the two following two cases appears

@. There and ?. , Let X i be the bag which contains both a and b. It is clear that ? ?1 (X i ) contains both x and

A. Edge, if e belongs to E(F ) We want to prove that there exists a matching in A covering all the vertices of F (G) Given a set of faces {F 1 , . . . , F i }, we need to prove that the corresponding set has at least i neighbours in A. Let us consider the planar graph S obtained by taking the union of F i 's, we have

]. H. Bodlaender and F. V. Fomin, As H is a forest, the number of edges of H incident to vertices of S is at most n S ? 1. So the hypothesis of Hall's theorem are satisfied. This proves that H is a nice subgraph Barnette proved in [1] the following theorem: References [1] D. Barnette. Trees in Polyhedral Graphs Approximation of Path-width of Outerplanar Graphs, Canadian Journal of Mathematics Journal of Algorithms, vol.43, issue.1822, pp.731-736190, 1966.

H. L. Bodlaender, Tree-width: Algorithmic Techniques and Results, Proceedings of 22nd International Symposium MFCS, p.1295, 1997.
DOI : 10.1007/bfb0029946
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.4015

V. Bouchitté, F. Mazoit, and I. Todinca, Chordal embeddings of planar graphs, Discrete Mathematics, vol.273, issue.1-3, pp.85-102, 2003.
DOI : 10.1016/S0012-365X(03)00230-9

D. Coudert, F. Huc, and J. S. Sereni, Pathwidth of outerplanar graphs, Journal of Graph Theory, vol.32, issue.1, pp.27-41, 2007.
DOI : 10.1002/jgt.20218
URL : https://hal.archives-ouvertes.fr/inria-00070220

A. Czumaj and W. B. Strothmann, Bounded degree spanning trees, Springer Lecture Notes in Computer Science 1284, pp.104-117, 1997.
DOI : 10.1007/3-540-63397-9_9

M. B. Dillencourt, Finding Hamiltonian cycles in Delaunay triangulations is NP-complete, Discrete Applied Mathematics, vol.64, issue.3, pp.207-217, 1996.
DOI : 10.1016/0166-218X(94)00125-W

F. V. Fomin and D. M. Thilikos, On self duality of pathwidth in polyhedral graph embeddings, Journal of Graph Theory, vol.14, issue.1, pp.42-45, 2007.
DOI : 10.1002/jgt.20219

D. Lapoire, Structuration des graphes planaires, 1999.

B. Reed, Tree Width and Tangles: A New Connectivity Measure and Some Applications, Surveys in Combinatorics, pp.87-162, 1997.
DOI : 10.1017/CBO9780511662119.006

N. Robertson and P. D. Seymour, Graph minors. I. Excluding a forest, Journal of Combinatorial Theory, Series B, vol.35, issue.1, pp.39-61, 1983.
DOI : 10.1016/0095-8956(83)90079-5

N. Robertson and P. D. Seymour, Graph minors. III. Planar tree-width, Journal of Combinatorial Theory, Series B, vol.36, issue.1, pp.49-64, 1984.
DOI : 10.1016/0095-8956(84)90013-3

P. Seymour and R. Thomas, Call routing and the ratcatcher, Combinatorica, vol.32, issue.2, pp.217-241, 1994.
DOI : 10.1007/BF01215352

T. T. Costa, L. Létocart, and F. Roupin, A Theorem on Planar Graphs Minimal multicut and maximal integer multiflow : A survey, Transactions of American Mathematical Society European Journal of Operational Research, vol.82, issue.1621, pp.99-11655, 1956.

D. Coudert and H. Rivano, Lightpath assignment for multifibers WDM networks with wavelength translators, Global Telecommunications Conference, 2002. GLOBECOM '02. IEEE, pp.2686-2690, 2002.
DOI : 10.1109/GLOCOM.2002.1189117
URL : https://hal.archives-ouvertes.fr/inria-00072101

H. [. Coudert, X. Rivano, and . Roche, A Combinatorial Approximation Algorithm for the Multicommodity Flow Problem, WAOA 03, number 2909 in Lecture Notes in Computer Science, pp.256-259, 2003.
DOI : 10.1007/978-3-540-24592-6_22
URL : https://hal.archives-ouvertes.fr/inria-00429176

J. [. Desaulniers, M. M. Desrosiers, and . Solomon, Column Generation, 2005.
DOI : 10.1007/b135457

L. K. Fleischer, Approximating Fractional Multicommodity Flow Independent of the Number of Commodities, SIAM Journal on Discrete Mathematics, vol.13, issue.4, pp.505-520, 2000.
DOI : 10.1137/S0895480199355754

V. [. Garg, M. Vazirani, and . Yannakakis, Approximate max-flow min(multi )cut theorems and their applications, ACM Symposium on Theory of Computing, pp.698-707, 1993.
DOI : 10.1137/s0097539793243016
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.3217

B. Jaumard, C. Meyer, and B. Thiongane, On column generation formulations for the RWA problem, International Network Optimization Conference (INOC'O5), pp.20-23, 2005.
DOI : 10.1016/j.dam.2008.08.033

]. P. Rag88 and . Raghavan, Probabilistic construction of deterministic algorithms : approximating packing integer programs, Journal of Computer and System Sciences, vol.37, issue.2, pp.130-143, 1988.

B. Ch, O. Amini, J. Bermond, F. Giroire, F. Huc et al., Design of minimal fault tolerant networks : Asymptotic bounds, Huitièmes Rencontres Francophones sur les Aspects Algorithmiques des Télécommunications (AlgoTel'06), pp.37-40, 2006.

N. Alon and M. Capalbo, Smaller Explicit Superconcentrators, Proceedings of the Fourteenth Annual Symposium on Discret Algorithms ACM-SIAM (SODA), pp.340-346, 2003.
DOI : 10.1080/15427951.2004.10129083

F. [. Amini, F. Giroire, S. Huc, and . Pérennes, Minimal selectors and fault tolerant networks, Networks, vol.98, issue.4
DOI : 10.1002/net.20326
URL : https://hal.archives-ouvertes.fr/inria-00082015

N. Alon, Z. Galil, and V. D. Milman, Better expanders and superconcentrators, Journal of Algorithms, vol.8, issue.3, pp.337-347, 1987.
DOI : 10.1016/0196-6774(87)90014-9

P. [. Alon, A. V. Hamburger, and . Kostochka, Regular Honest Graphs, Isoperimetric Numbers, and Bisection of Weighted Graphs, European Journal of Combinatorics, vol.20, issue.6, 1999.
DOI : 10.1006/eujc.1998.0295

A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, 1974.

]. N. Alo86 and . Alon, Eigenvalues and expanders [Alo93] N. Alon. On the edge-expansion of graphs, Combinatorica Combinatorics, Probability and Computing, vol.6, issue.11, pp.83-961, 1986.

N. Alon and V. D. Milman, Eigenvalues, Expanders And Superconcentrators, 25th Annual Symposium onFoundations of Computer Science, 1984., pp.320-322, 1984.
DOI : 10.1109/SFCS.1984.715931
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.300.4326

J. Bermond, E. Darrot, and O. Delmas, Design of fault-tolerant networks for satellites (TWTA redundancy), Networks, vol.40, issue.4, pp.202-207, 2002.
DOI : 10.1002/net.10044
URL : https://hal.archives-ouvertes.fr/hal-00307611

J. Bermond, F. Giroire, and S. Pérennes, Design of Minimal Fault Tolerant On-Board Networks: Practical Constructions, 14th International Colloquium on Structural Information and Communication Complexity, pp.261-273, 2007.
DOI : 10.1007/978-3-540-72951-8_21
URL : https://hal.archives-ouvertes.fr/hal-00512282

J. Bermond, F. Havet, and D. Tóth, Fault tolerant on-board networks with priorities, Networks, vol.6, issue.1, pp.9-25, 2006.
DOI : 10.1002/net.20094
URL : https://hal.archives-ouvertes.fr/inria-00070640

Y. Bilu and N. Linial, Lifts, Discrepancy and Nearly Optimal Spectral Gap*, Combinatorica, vol.26, issue.5, pp.495-519, 2006.
DOI : 10.1007/s00493-006-0029-7

B. Bollobás, The Isoperimetric Number of Random Regular Graphs, European Journal of Combinatorics, vol.9, issue.3, pp.241-244, 1988.
DOI : 10.1016/S0195-6698(88)80014-3

F. R. Chung, M. Capalbo, O. Reingold, S. Vadhan, and A. Wigderson, Spectral graph theory Regional conference series in mathematics Randomness conductors and constant-degree lossless expanders, 34th ACM Symposium on Theory of Computing (STOCS), pp.659-668, 1997.

]. E. Dar97 and . Darrot, Diffusion de messages longs et propriétés structurelles dans les réseaux d'interconnexion, 1997.

F. [. Delmas, M. Havet, S. Montassier, and . Pérennes, Design of fault tolerant on-board networks, INRIA Research Report, vol.5866, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01111370

[. Du and H. Q. Ngo, Notes on the complexity of switching networks, pp.307-367, 2001.

G. Davidovf, P. Sarnak, and A. Valette, Elementary Number Theory, Group Theory, and Ramanujan Graphs, 2003.

K. Y. Eng and M. Hecht, Switch matrix for TWTA redundancy on communication satellites, National Telecommunications Conference (NTC'78), 1978.

]. F. Gir06 and . Giroire, Réseaux, algorithmique et analyse combinatoire de grands ensembles, 2006.

]. F. Hav06 and . Havet, Repartitors, selectors and superselectors, Journal of Interconnexion Networks, vol.7, issue.3, pp.391-415, 2006.

G. Lev and L. G. Valiant, Size bounds for superconcentrators, Theoretical Computer Science, vol.22, issue.3, pp.233-251, 1983.
DOI : 10.1016/0304-3975(83)90105-6

B. Monien and R. Preis, Upper bounds on the bisection width of 3- and 4-regular graphs, Journal of Discrete Algorithms, vol.4, issue.3, pp.475-498, 2006.
DOI : 10.1016/j.jda.2005.12.009

M. Pinsker, On the complexity of a concentrator, 7th International Teletraffic Conference, p.318, 1973.

L. G. Valiant, On non-linear lower bounds in computational complexity, Proceedings of seventh annual ACM symposium on Theory of computing , STOC '75, pp.45-53, 1975.
DOI : 10.1145/800116.803752

B. Ch, S. Alouf, E. Altman, J. Galtier, J. F. Lalande et al., Algorithmes de routage An algorithm for satellite bandwidth allocation, Proceedings of IEEE infocom 2005, pp.560-571, 2005.

E. S. Alouf, J. Altman, J. F. Galtier, C. Lalande, and . Touati, Quasi-Optimal Resource Allocation in Multi-Spot MFTDMA Satellite Networks, Combinatorial Optimization in Communication Networks, 2006.
URL : https://hal.archives-ouvertes.fr/inria-00000005

O. Amini, F. Huc, I. S. Valls, and J. Zerovnik, (?, k)-Routing on Plane Grids. ArXiv e-prints, 2008.

X. [. Akyildiz, W. Wang, and . Wang, Wireless mesh networks: a survey, Computer Networks, vol.47, issue.4, pp.445-487, 2005.
DOI : 10.1016/j.comnet.2004.12.001

J. Bermond, F. Havet, F. Huc, and C. Linhares, Allocation de fréquence et coloration impropre des graphes hexagonaux pondérés Ile d'Oléron, Neuvièmes Rencontres Francophones sur les Aspects Algorithmiques des Télécommunications (Al- goTel'07), 2007.

. B. Cbf-+-03-]-j, T. Cain, L. Billhartz, E. Foore, J. Althouse et al., A link scheduling and ad hoc networking approach using directional antennas, IEEE Military Communications Conference, vol.1, 2003.

R. [. Havet, J. S. Kang, and . Sereni, Improper Colourings of Unit Disk Graphs, Electronic Notes in Discrete Mathematics, vol.22, pp.123-128, 2005.
DOI : 10.1016/j.endm.2005.06.022

T. [. Hwang, R. H. Lin, and . Jan, A Permutation Routing Algorithm for Double Loop Networks, Parallel Processing Letters, vol.07, issue.03, pp.259-265, 1997.
DOI : 10.1142/S0129626497000279

F. Huc, C. Linhares, and H. Rivano, The proportional colouring problem : Optimizing buffers in radio mesh networks, IV Latin-American Algorithms, Graphs and Optimization Symposium (LAGOS'07, pp.25-29, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00429822

N. [. Klasing, S. Morales, and . Pérennes, On the complexity of bandwidth allocation in radio networks with steady traffic demands. Theoretical Computer Science
URL : https://hal.archives-ouvertes.fr/inria-00070575

J. Lalande, Conception de réseaux de télécommunications : optimisation et expérimentations, 2004.

T. Leighton, B. Maggs, and S. Rao, Universal packet routing algorithms, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science, pp.256-269, 1988.
DOI : 10.1109/SFCS.1988.21942
URL : http://www.dtic.mil/get-tr-doc/pdf?AD=ADA204273

B. [. Leighton, S. B. Maggs, . [. Rao, B. Liu, and . Zhao, Packet routing and job-shop scheduling inO(congestion+dilation) steps, International Conference on Computer Networks and Mobile Computing, pp.167-186, 1994.
DOI : 10.1007/BF01215349
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.151.8752

C. Mcdiarmid and B. Reed, Channel assignment and weighted coloring, Networks, vol.29, issue.2, pp.114-117, 2000.
DOI : 10.1002/1097-0037(200009)36:2<114::AID-NET6>3.0.CO;2-G

L. Narayanan and S. M. Shende, Static Frequency Assignment in Cellular Networks, Algorithmica, vol.29, issue.3, pp.396-409, 2001.
DOI : 10.1007/s004530010067

A. Pietracaprina and G. Pucci, Optimal Many-to-One Routing on the Mesh with Constant Queues, Lecture Notes in Computer Science, vol.2150, pp.645-649, 2001.
DOI : 10.1007/3-540-44681-8_92

J. F. Sibeyn, B. S. Chlebus, and M. Kaufmann, Deterministic Permutation Routing on Meshes, Journal of Algorithms, vol.22, issue.1, pp.111-141, 1997.
DOI : 10.1006/jagm.1995.0804

J. S. Sereni, Coloration de graphes et applications, 2006.

J. F. Sibeyn and M. Kaufman, Deterministic 1 -k routing on meshes with applications to worm-hole routing, LNCS, editor, 11th Symposium on Theoretical Aspects of Computer Science, pp.237-248, 1994.
DOI : 10.1007/3-540-57785-8_145

A. Srinivasan and C. Teo, A constant-factor approximation algorithm for packet routing, and balancing local vs. global criteria, Proceedings of the twentyninth annual ACM symposium on Theory of computing (STOC'97), pp.636-643, 1997.

. Ch, Réseaux d'accès et de coeur [AA89] I. Algor and N. Alon. The star arboricity of graphs, Discrete Mathematics, vol.5, issue.751-3, pp.11-22, 1989.

F. [. Amini, F. Havet, S. Huc, and . Thomassé, WDM and Directed Star Arboricity, Combinatorics, Probability and Computing, vol.41, issue.02, 2007.
DOI : 10.1016/0012-365X(89)90073-3
URL : https://hal.archives-ouvertes.fr/inria-00132396

V. Alwayn, Fiber-optic technologies, 2004.

A. Boudani, A. Guitton, and B. Cousin, GXcast: generalized explicit multicast routing protocol, Proceedings. ISCC 2004. Ninth International Symposium on Computers And Communications (IEEE Cat. No.04TH8769), 2004.
DOI : 10.1109/ISCC.2004.1358673
URL : https://hal.archives-ouvertes.fr/hal-01503027

. Vance, Branch-and-price : Column generation for solving huge integer programs, Operations Research, vol.46, issue.3, pp.316-329, 1998.

]. R. Bibliographie-[-bra03 and . Brandt, Multicasting using wdm in multfiber optical star networks, 2003.

C. C. Cordatcou01 and ]. Coudert, La fibre optique Algorithmique et optimisation de réseaux de communications optiques, Dee91] S. Deering. Multicast Routing in a Datagram Network, 1991.

B. Guiduli, On incidence coloring and star arboricity of graphs, Discrete Mathematics, vol.163, issue.1-3, pp.275-278, 1997.
DOI : 10.1016/0012-365X(95)00342-T

A. Guitton, Communications multicast, Contributions aux réseaux optiques et au passage à l'échelle, 2005.

J. He, S. H. Chan, and D. H. Tsang, Routing and wavelength assignment for WDM multicast networks, IEEE Global Telecommunications Conference (GLOBECOM'01, 2001.

B. Jaumard, C. Meyer, and B. Thiongane, ILP formulations for the RWA problem for symmetrical systems, Handbook for Optimization in Telecommunications, chapter 23, pp.637-678, 2006.

B. [. Mukherjee, X. Malli, C. Zhang, and . Qiao, Benefits of Multicasting in All-Optical Networks, Optical WDM Networks, pp.209-220, 1998.

A. Pinlou and É. Sopena, The acircuitic directed star arboricity of subcubic graphs is at most four, Discrete Mathematics, vol.306, issue.24, pp.3281-3289, 2006.
DOI : 10.1016/j.disc.2006.06.007
URL : https://hal.archives-ouvertes.fr/hal-00307095

G. N. Rouskas, Optical layer multicast: rationale, building blocks, and challenges, IEEE Network, vol.17, issue.1, pp.60-65, 2003.
DOI : 10.1109/MNET.2003.1174179

R. Ramaswami and K. N. Sivarajan, Optical Networks : A Practical Perspective, 2002.

J. Siuzdak, ISI and polarization switching in high bit rate optical fiber transmission systems, Optical and Quantum Electronics, vol.35, issue.12, pp.1113-1121, 2003.
DOI : 10.1023/A:1026208519556

M. K. Shin, Y. J. Kim, K. S. Park, and S. H. Kim, Explicit Multicast Extension (Xcast+) for Efficient Multicast Packet Delivery, ETRI Journal, vol.23, issue.4, pp.202-204, 2001.
DOI : 10.4218/etrij.01.0201.0402

L. H. Sahasrabuddhe and B. Mukherjee, Light trees: optical multicasting for improved performance in wavelength routed networks, IEEE Communications Magazine, vol.37, issue.2, pp.67-73, 1999.
DOI : 10.1109/35.747251

. Ch, Fiabilité et tolérance aux pannes [Bou05] A. Bouffard. Dimensionnement GRWA et protection par segment dans les réseaux optiques WDM, Canada, vol.6, 2005.

]. D. Cdp-+-07, P. Coudert, S. Datta, H. Perennes, M. Rivano et al., Shared risk resource group : Complexity and approximability issues, Parallel Processing Letters, vol.17, issue.2, pp.169-184, 2007.

D. Coudert, F. Huc, F. Peix, and M. Voge, Reliability of Connections in Multilayer Networks under Shared Risk Groups and Costs Constraints, 2008 IEEE International Conference on Communications, pp.5170-5174, 2008.
DOI : 10.1109/ICC.2008.971
URL : https://hal.archives-ouvertes.fr/inria-00429150

D. Coudert, S. Pérennes, H. Rivano, and M. Voge, Shared Risk Resource Groups and Survivability in Multilayer Networks, 2006 International Conference on Transparent Optical Networks, pp.235-238, 2006.
DOI : 10.1109/ICTON.2006.248442
URL : https://hal.archives-ouvertes.fr/inria-00429170

G. [. Dutta and . Rouskas, Traffic grooming in WDM networks: past and future, IEEE Network, vol.16, issue.6, pp.46-56, 2002.
DOI : 10.1109/MNET.2002.1081765

R. [. Huang, G. N. Dutta, and . Rouskas, Traffic grooming in path, star, and tree networks: complexity, bounds, and algorithms, IEEE Journal on Selected Areas in Communications, vol.24, issue.4, 2006.
DOI : 10.1109/JSAC.2006.1613773

J. Q. Hu and B. Leida, Traffic grooming, routing, and wavelength assignment in optical WDM mesh networks, IEEE INFOCOM 2004, pp.495-501, 2004.
DOI : 10.1109/INFCOM.2004.1354521

[. Ho and H. T. Mouftah, A framework for service-guaranteed shared protection in wdm mesh networks, IEEE Communications Magazine, pp.97-103, 2002.

E. Modiano and P. Lin, Traffic grooming in WDM networks, IEEE Communications Magazine, vol.39, issue.7, pp.124-129, 2001.
DOI : 10.1109/35.933446

T. Noronha and C. Ribeiro, Routing and wavelength assignment by partition coloring, European Journal of Operational Research, vol.1713, issue.3, pp.797-810, 2006.
DOI : 10.1016/j.ejor.2004.09.007

M. Pioro and D. Medhi, Routing, flow, and capacity design in communication and computer networks, 2004.

]. A. Som06 and . Somani, Survivability and Traffic Grooming in WDM Optical Networks, 2006.

M. Voge, How to Transform a Multilayer Network into a Colored Graph, 2006 International Conference on Transparent Optical Networks, pp.116-119, 2006.
DOI : 10.1109/ICTON.2006.248415

M. Voge, Optimisation des réseaux de télécommunications : réseaux multicouches , tolérance aux pannes et surveillance de trafic, 2006.

Y. S. Bibliographie, S. Yuan, J. P. Varma, and . Jue, Minimum-color path problems for reliability in mesh networks, IEEE INFOCOM, pp.2658-2669, 2005.

K. Zhu and B. Mukherjee, A review of traffic grooming in WDM optical networks : Architectures and challenges, Optical Networks Magazine, pp.55-64, 2003.

]. O. Chahp, F. Amini, S. Huc, and . Pérennes, Reroutage de connexions et Pathwidth On the pathwidth of planar graphs, SIAM Journal of Discrete Mathematics, vol.7

H. L. Bodlaender and F. V. Fomin, Approximation of pathwidth of outerplanar graphs, Journal of Algorithms, vol.43, issue.2, pp.190-200, 2002.
DOI : 10.1016/S0196-6774(02)00001-9

L. Barrière, P. Flocchini, P. Fraigniaud, and N. Santoro, Capture of an intruder by mobile agents, Proceedings of the fourteenth annual ACM symposium on Parallel algorithms and architectures , SPAA '02, pp.200-209, 2002.
DOI : 10.1145/564870.564906

H. L. Bodlaender and T. Kloks, Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs, Journal of Algorithms, vol.21, issue.2, pp.358-402, 1996.
DOI : 10.1006/jagm.1996.0049

]. D. Chm08a, F. Coudert, D. Huc, and . Mazauric, Algorithme générique pour les jeux de capture dans les arbres, Dixième Rencontres Francophones sur les Aspects Algorithmiques des Télécommunications (AlgoTel'08), 2008.

D. Coudert, F. Huc, and D. Mazauric, A Distributed Algorithm for Computing and Updating the Process Number of a Forest, Research Report, vol.6560, p.6, 2008.
DOI : 10.1007/978-3-540-87779-0_36
URL : https://hal.archives-ouvertes.fr/inria-00373850

]. D. Chm08c, F. Coudert, D. Huc, and . Mazauric, A distributed algorithm for computing and updating the process number of a forest (brief announcement), 22nd International Symposium on Distributed Computing (DISC), 2008.

F. [. Coudert, J. Huc, and . Sereni, Pathwidth of outerplanar graphs, Journal of Graph Theory, vol.32, issue.1, pp.27-41, 2007.
DOI : 10.1002/jgt.20218
URL : https://hal.archives-ouvertes.fr/inria-00070220

D. Coudert, S. Pérennes, Q. C. Pham, and J. Sereni, Rerouting requests in WDM networks, Septièmes rencontres francopohones sur les Aspects Algorithmiques des Télécommunications (AlgoTel'05), 2005.
URL : https://hal.archives-ouvertes.fr/inria-00429173

J. [. Coudert and . Sereni, Characterization of graphs and digraphs with small process numbers, Discrete Applied Mathematics, vol.159, issue.11, 2007.
DOI : 10.1016/j.dam.2011.03.010

J. A. Ellis, I. H. Sudborough, and J. Turner, The Vertex Separation and Search Number of a Graph, Information and Computation, vol.113, issue.1, pp.50-79, 1994.
DOI : 10.1006/inco.1994.1064

]. F. Fom03 and . Fomin, Pathwidth of planar and line graphs, Graphs and Combinatorics, vol.19, issue.1, pp.91-99, 2003.

D. [. Fomin and . Thilikos, On self-duality for pathwidth in polyhedral graph embeddings Fomin and D.M. Thilikos. An annotated bibliography on guaranteed graph searching, Journal of Graph Theory Theor. Comput. Sci, vol.55, issue.3993, pp.236-245, 2007.

C. [. Kirousis and . Papadimitriou, Searching and pebbling, Theoretical Computer Science, vol.47, issue.2, pp.205-218, 1986.
DOI : 10.1016/0304-3975(86)90146-5
URL : http://doi.org/10.1016/0304-3975(86)90146-5

]. A. Lap93 and . Lapaugh, Recontamination does not help to search a graph, Journal of the Association for Computing Machinery (ACM), vol.40, issue.2, pp.224-245, 1993.

D. Lapoire, Structuration des graphes planaires, 1999.

]. N. Nis07 and . Nisse, Les jeux des gendarmes et du voleur dans les graphes. Théorie des mineurs, stratégies connexes et approche distribuée, 2007.

Q. C. Pham, Étude d'un problème algorithmique intervenant dans le routage optique semi-dynamique, 2004.

B. Reed, Treewidth and tangles : a new connectivity measure and some applications, Surveys in Combinatorics, pp.87-162, 1997.

N. Robertson and P. D. Seymour, Graph minors. I. Excluding a forest, Journal of Combinatorial Theory, Series B, vol.35, issue.1, pp.39-61, 1983.
DOI : 10.1016/0095-8956(83)90079-5

K. Skodinis, Computing Optimal Linear Layouts of Trees in Linear Time, 8th Annual European Symposium, 2000.
DOI : 10.1007/3-540-45253-2_37

A. Amini, S. Griffiths, and F. Huc, 4-cycles in mixing digraphs, Electronic Notes in Discrete Mathematics, vol.30, issue.30, pp.63-68, 2008.
DOI : 10.1016/j.endm.2008.01.012

A. [. Alon and . Shapira, Testing subgraphs in directed graphs, Journal of Computer and System Sciences, vol.69, issue.3, pp.354-382, 2004.
DOI : 10.1016/j.jcss.2004.04.008

P. [. Brown, M. Erdos, and . Simonovits, Extremal problems for directed graphs, Journal of Combinatorial Theory, Series B, vol.15, issue.1, pp.77-93, 1973.
DOI : 10.1016/0095-8956(73)90034-8

W. G. Brown, P. Erdos, and M. Simonovits, Algorithmic solution of extremal digraph problems. Transactions of the, pp.421-449, 1985.

J. A. Bondy and M. Simonovits, Cycles of even length in graphs, Journal of Combinatorial Theory, Series B, vol.16, issue.2, pp.97-105, 1974.
DOI : 10.1016/0095-8956(74)90052-5

. [. Bibliographie, A. Erd?s, V. T. Rényi, and . Sós, A problem in Graph Theory, American Mathematical Monthly, vol.71, pp.1107-1110, 1964.

P. Erd?s and M. Simonovits, A limit theorem in graph theory, Studia Scientiarum Mathematicarum Hungarica, vol.1, pp.51-5751, 1966.

P. Erd?s and M. Simonovits, Compactness results in extremal graph theory, Combinatorica, vol.66, issue.3, pp.275-288, 1982.
DOI : 10.1007/BF02579234

Z. Furedi, Graphs without quadrilaterals, Journal of Combinatorial Theory, Series B, vol.34, issue.2, pp.187-190, 1983.
DOI : 10.1016/0095-8956(83)90018-7

]. S. Gri07 and . Griffiths, One way cuts in oriented graphs, 2007.

J. Håstad, Clique is hard to approximate within n1?????, Acta Mathematica, vol.182, issue.1, pp.105-142, 1999.
DOI : 10.1007/BF02392825

J. Komlós, A. Shokoufandeh, M. Simonovits, and E. Szemerédi, The regularity lemma and its applications in graph theory. Theoretical aspects of computer science : advanced lectures, pp.84-112, 2002.

V. [. Lazebnik, A. J. Ustimenko, and . Woldar, A New Series of Dense Graphs of High Girth, Bulletin of the American Mathematical Society, vol.32, issue.1, pp.73-79, 1995.
DOI : 10.1090/S0273-0979-1995-00569-0

T. Lam and J. Verstraete, A note on graphs without short even cycles, The Electronic Journal of Combinatorics, vol.12, 2005.

J. [. Naor and . Verstraete, On the Turan number for the hexagon, Advances in Mathematics, vol.203, issue.2, pp.476-496, 2006.

]. E. Sze75 and . Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica, vol.27, pp.199-245, 1975.