. Proof, As in A, called avenue, in T , such that all connected components S of T \ A are such that ws(S) < k. By the assumption, for any v ? V (T ), there is no branch R at v with ws(R) > k and there are at most 2 branches R at v with ws(R) = k, Let v ? V (T ) that has the maximum number of branches S with ws(S) = k

]. D. Bienstock, Graph searching, path-width, tree-width and related problems (a survey), DIMACS Ser. in Discrete Mathematics and Theoretical Computer Science, vol.5, pp.33-49, 1991.
DOI : 10.1090/dimacs/005/02

D. Bienstock and P. D. Seymour, Monotonicity in graph searching, Journal of Algorithms, vol.12, issue.2, pp.239-245, 1991.
DOI : 10.1016/0196-6774(91)90003-H

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

J. R. Blair, F. Manne, and R. Mihai, Efficient Self-stabilizing Graph Searching in Tree Networks, Proceedings of 12th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS), pp.111-125, 2010.
DOI : 10.1007/978-3-642-16023-3_11

L. Blin, P. Fraigniaud, N. Nisse, and S. Vial, Distributed chasing of network intruders, Theoretical Computer Science, vol.399, issue.1-2, pp.12-37, 2008.
DOI : 10.1016/j.tcs.2008.02.004

URL : https://hal.archives-ouvertes.fr/hal-01310347

L. Blin, A. Milani, M. Potop-butucaru, and S. Tixeuil, Exclusive Perpetual Ring Exploration without Chirality, 24th International Symposium on Distributed Computing (DISC), pp.312-327, 2010.
DOI : 10.1007/978-3-642-15763-9_29

URL : https://hal.archives-ouvertes.fr/hal-00992700

F. Bonnet, A. Milani, M. Potop-butucaru, and S. Tixeuil, Asynchronous Exclusive Perpetual Grid Exploration without Sense of Direction, OPODIS, pp.251-265, 2011.
DOI : 10.1007/978-3-642-25873-2_18

URL : https://hal.archives-ouvertes.fr/hal-00992679

R. L. Breisch, An intuitive approach to speleotopology, Southwestern Cavers, vol.6, pp.72-78, 1967.

R. L. Breisch, Lost in a Cave-applying graph theory to cave exploration, 2012.

J. Chalopin, P. Flocchini, B. Mans, and N. Santoro, Network Exploration by Silent and Oblivious Robots, 36th International Workshop on Graph Theoretic Concepts in Computer Science (WG), pp.208-219, 2010.
DOI : 10.1137/S009753979628292X

URL : https://hal.archives-ouvertes.fr/hal-01198880

D. Coudert, F. Huc, and D. Mazauric, A Distributed Algorithm for Computing and Updating the Process Number of a Forest, Algorithmica, pp.453-464, 2011.
DOI : 10.1007/978-3-540-87779-0_36

URL : https://hal.archives-ouvertes.fr/inria-00373850

G. D. Angelo, G. D. Stefano, and A. Navarra, Gathering of six robots on anonymous symmetric rings, SIROCCO, pp.174-185, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00644039

N. Deo, M. S. Krishnamoorthy, and M. A. Langston, Exact and Approximate Solutions for the Gate Matrix Layout Problem, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol.6, issue.1, pp.79-84, 1987.
DOI : 10.1109/TCAD.1987.1270248

E. W. Dijkstra, Self-stabilizing systems in spite of distributed control, Communications of the ACM, vol.17, issue.11, pp.643-644, 1974.
DOI : 10.1145/361179.361202

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

P. Flocchini, D. Ilcinkas, A. Pelc, and N. Santoro, Computing Without Communicating: Ring Exploration by Asynchronous Oblivious Robots, 11th Int. Conf. on Princ. of Dist. Syst. (OPODIS), pp.105-118, 2007.
DOI : 10.1007/978-3-540-77096-1_8

URL : https://hal.archives-ouvertes.fr/hal-00339884

P. Flocchini, D. Ilcinkas, A. Pelc, and N. Santoro, Remembering without memory: Tree exploration by asynchronous oblivious robots, Theor. Comput. Sci, vol.411, pp.14-151583, 2010.
DOI : 10.1016/j.tcs.2010.01.007

URL : https://hal.archives-ouvertes.fr/hal-00341465

P. Flocchini, D. Ilcinkas, A. Pelc, and N. Santoro, How many oblivious robots can explore a line, Information Processing Letters, vol.111, issue.20, pp.1027-1031, 2011.
DOI : 10.1016/j.ipl.2011.07.018

URL : https://hal.archives-ouvertes.fr/hal-00643668

P. Flocchini, G. Prencipe, N. Santoro, and P. Widmayer, Distributed coordination of a set of autonomous mobile robots, Proceedings of the IEEE Intelligent Vehicles Symposium 2000 (Cat. No.00TH8511), pp.480-485, 2000.
DOI : 10.1109/IVS.2000.898389

F. V. Fomin and D. M. Thilikos, An annotated bibliography on guaranteed graph searching, Theoretical Computer Science, vol.399, issue.3, pp.236-245, 2008.
DOI : 10.1016/j.tcs.2008.02.040

D. Ilcinkas, N. Nisse, and D. Soguet, The cost of monotonicity in distributed graph searching, Distributed Computing, vol.410, issue.14, pp.117-127, 2009.
DOI : 10.1007/s00446-009-0089-1

URL : https://hal.archives-ouvertes.fr/hal-00412063

L. 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

URL : http://doi.org/10.1016/0304-3975(86)90146-5

R. Klasing, A. Kosowski, and A. Navarra, Taking advantage of symmetries: Gathering of many asynchronous oblivious robots on a ring, Theoretical Computer Science, vol.411, issue.34-36, pp.34-363235, 2010.
DOI : 10.1016/j.tcs.2010.05.020

URL : https://hal.archives-ouvertes.fr/hal-00519069

R. Klasing, E. Markou, and A. Pelc, Gathering asynchronous oblivious mobile robots in a ring, Theoretical Computer Science, vol.390, issue.1, pp.27-39, 2008.
DOI : 10.1016/j.tcs.2007.09.032

URL : https://hal.archives-ouvertes.fr/hal-00307234

A. S. Lapaugh, Recontamination does not help to search a graph, Journal of the ACM, vol.40, issue.2, pp.224-245, 1993.
DOI : 10.1145/151261.151263

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

R. Mihai and M. Mjelde, A Self-stabilizing Algorithm for Graph Searching in Trees, Proceedings of 11th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS), pp.563-577, 2009.
DOI : 10.1007/978-3-642-05118-0_39

B. Monien and I. H. Sudborough, Min cut is NP-complete for edge weighted trees, Theoretical Computer Science, vol.58, issue.1-3, pp.209-229, 1988.
DOI : 10.1016/0304-3975(88)90028-X

N. Nisse and D. Soguet, Graph searching with advice, Theoretical Computer Science, vol.410, issue.14, pp.1307-1318, 2009.
DOI : 10.1016/j.tcs.2008.08.020

URL : https://hal.archives-ouvertes.fr/inria-00423450

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

G. Prencipe, Instantaneous Actions vs. Full Asynchronicity: Controlling and Coordinating a Sset of Autonomous Mobile Robots, ICTCS, pp.154-171, 2001.
DOI : 10.1007/3-540-45446-2_10

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

K. Skodinis, Computing Optimal Linear Layouts of Trees in Linear Time, J. Algorithms, vol.47, issue.1, pp.40-59, 2003.
DOI : 10.1007/3-540-45253-2_37