. Symmetric, then there is another hole which size is more than 2, it is a contradiction If the destination of P in Alg1 is to B1 and the size of B1 is equal to 1, then B1 moves to B0, the size of B2 is 2, and the robot

P. If, after R joins B0, then new destination of P is to B0 by both of Alg1 and Alg2, Therefore, the system achieves the gathering in O(k 2 ) rounds

L. Blin, A. Milani, M. Potop-butucaru, and S. Tixeuil, Exclusive Perpetual Ring Exploration without Chirality, DISC, pp.312-327, 2010.
DOI : 10.1007/978-3-642-15763-9_29
URL : https://hal.archives-ouvertes.fr/hal-00992700

M. Cieliebak, Gathering Non-oblivious Mobile Robots, LATIN, pp.577-588, 2004.
DOI : 10.1007/978-3-540-24698-5_60

M. Cieliebak, P. Flocchini, G. Prencipe, and N. Santoro, Solving the Robots Gathering Problem, ICALP, pp.1181-1196, 2003.
DOI : 10.1007/3-540-45061-0_90

A. Dessmark, P. Fraigniaud, D. R. Kowalski, and A. Pelc, Deterministic Rendezvous in Graphs, Algorithmica, vol.46, issue.1, pp.69-96, 2006.
DOI : 10.1007/s00453-006-0074-2

S. Devismes, F. Petit, and S. Tixeuil, Optimal probabilistic ring exploration by semi-synchronous oblivious robots, SIROCCO, pp.195-208, 2009.
DOI : 10.1007/978-3-642-11476-2_16
URL : https://hal.archives-ouvertes.fr/hal-00930045

P. Flocchini, D. Ilcinkas, A. Pelc, and N. Santoro, Computing without communicating: Ring exploration by asynchronous oblivious robots, OPODIS, pp.105-118, 2007.
DOI : 10.1007/978-3-540-77096-1_8
URL : https://hal.archives-ouvertes.fr/hal-00339884

P. Flocchini, E. Kranakis, D. Krizanc, N. Santoro, and C. Sawchuk, Multiple Mobile Agent Rendezvous in a Ring, LATIN, pp.599-608, 2004.
DOI : 10.1007/978-3-540-24698-5_62

P. Flocchini, G. Prencipe, N. Santoro, and P. Widmayer, Gathering of asynchronous robots with limited visibility, Theoretical Computer Science, vol.337, issue.1-3, pp.147-168, 2005.
DOI : 10.1016/j.tcs.2005.01.001

T. Izumi, T. Izumi, S. Kamei, and F. Ooshita, Mobile Robots Gathering Algorithm with Local Weak Multiplicity in Rings, SIROCCO, pp.101-113, 2010.
DOI : 10.1007/978-3-642-13284-1_9

R. Klasing, A. Kosowski, and A. Navarra, Taking advantage of symmetries: Gathering of asynchronous oblivious robots on a ring, OPODIS, pp.446-462, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00342931

R. Klasing, E. Markou, and A. Pelc, Gathering asynchronous oblivious mobile robots in a ring, ISAAC, pp.744-753, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00307234

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

R. Dariusz, A. Kowalski, and . Pelc, Polynomial deterministic rendezvous in arbitrary graphs, ISAAC, pp.644-656, 2004.

A. Lamani, M. Gradinariu-potop-butucaru, and S. Tixeuil, Optimal Deterministic Ring Exploration with Oblivious Asynchronous Robots, SIROCCO, pp.183-196, 2010.
DOI : 10.1007/978-3-642-13284-1_15
URL : https://hal.archives-ouvertes.fr/hal-01009453

L. Gianluca-de-marco, E. Gargano, D. Kranakis, A. Krizanc, U. Pelc et al., Asynchronous deterministic rendezvous in graphs, Theoretical Computer Science, vol.355, issue.3, pp.315-326, 2006.
DOI : 10.1016/j.tcs.2005.12.016

G. Prencipe, CORDA: Distributed coordination of a set of autonomous mobile robots, Proc. 4th European Research Seminar on Advances in Distributed Systems (ERSADS'01), pp.185-190, 2001.

G. Prencipe, On the Feasibility of Gathering by Autonomous Mobile Robots, SIROCCO, pp.246-261, 2005.
DOI : 10.1007/11429647_20