, size in the subgraph induced by N i in linear time Notice that S i is actually an independent set of maximum size in the subgraph induced by N i since any vertex of N i may only have at most one neighbor in N i since G is a cactus. Let S be one of the sets S i , i = 1, . . . , n, of maximum size. By construction, S is an independent 2-clique of maximum size among all independent 2-cliques in G in which every vertices have a common neighbor in V . Moreover, S can be found in linear time. Now we prove that S is an independent 2-clique of maximum size in G. If |S| ? 3, by Lemma 7.10 we know that ? =2 (G) ? 3. Indeed, if ? =2 (G) ? 4 then G would not be a cactus since S is an independent 2-clique of maximum size among all independent 2-cliques in G

, any independent 2-clique of size 3 would be included in an induced cycle C 6 (as in Figure 7.3) without chord (since G is a cactus) and we note such a cycle c. Since G is not isomorphic to a cycle of length 6, one of the vertices (say v) of the cycle c has a neighbor out of c. Then since G is a cactus, the two neighbors of v in c and one neighbor of v in V \ c is an independent 2-clique of size 3, which is a contradiction since |S| = 2. Thus, S is an independent 2-clique of maximum size, If |S| ?, vol.310, issue.4 7 2

, Other graph classes in which both problems are polynomialtime solvable

, Clique and Max Independent Set are polynomial-time solvable We first investigate a subclass of split graphs, namely threshold graphs. It follows from the definitions that a threshold graph G = (V, E) is a split graph with the following property: the vertices of the independent set S can be ordered as v 1, such that N G (v 1 ) ? N G (v 2 ) ? . . . ? N G (v p ). We denote by u 1 , . . . , u q the vertices of the clique K, and we suppose that d G (u 1 ) ? d G (u 2 ) ? . . . ? d G (u q ). Without loss of generality, we assume that there is no isolated vertex in G. Note that a threshold graph can be recognized in linear time

, Proposition 7.13. Max Independent 2-Clique is linear-time solvable on threshold graphs. Moreover, every threshold graph G without isolated vertices we have ? =2 (G) = ?(G)

, ) be a threshold graph with the previous decomposition into S and K

N. Let and . {u-r,-u-r+1, u q }, for some r ? 1. Then a maximum independent 2-clique in G is S if K \ N G (v p ) = ?, and otherwise it is S ? {z} with any z ? K \ N G (v p ), since in both cases the common neighbor of all these vertices is u q . Since Max Independent 4.1 Two different 2-partitions for the same graph (given by the black and white colors) in which each part has at least 2 vertices, the first partition, x does not satisfy the proportion condition of a community structure since the proportion of neighbors in the white part is 1

, but the proportion of neighbors in the black partition is 2

, The second partition gives a 2-community structure, p.58

.. , A weak 2-community structure of a graph (presented by the colors black and white) in which the vertex v does not satisfy the proportion condition of a 2-community structure but satisfies the weak proportion condition of a weak 2-community structure from Definition 4, p.59

W. , A complete graph in which a 3-community structure is given by the colors black, gray, p.62

.. , A disconnected graph with an isolated vertex in which there is no community structure, p.63

, Applying one step in Case 2(A) on the gray vertex decreases the size of the cut by one and creates two vertices in C 1 with 3 in-neighbors, p.68

.. , Splitting C 2 when |N | = 4 (vertices in N are in gray), p.72

.. , A 2-community structure {C 1 , C 2 } (C 1 in white, C 2 in black) for other graphs on 5 vertices, p.76

.. , 2-community structure in a graph of maximum degree 3. Candidates vertices to be moved are in gray, p.77

, An example of a graph in which all 2-community structures are balanced, p.88

.. , A cross gadget and a graph of maximum degree 3 without balanced 2- community structure, p.88

.. , A tree of maximum degree 3 in which any balanced 2-community structure (or even balanced weak 2-community structure) is disconnected (an example of a balanced 2-community structure is presented by the black and white colors), p.89

, 12 A graph with 10 vertices that does not contain any 2-community structure 90

G. , A schematic representation of a graph in, p.91

?. , , p.100

, A cubic graph where the community of maximum size (in gray) is disconnected, p.102

A. , T. , and E. , constructed by two stars S d for some integer d such that their center are joined by a path of length 2. The community in gray is a disconnected community of maximum size, whereas a connected community of maximum size have no more than |V | 2 vertices, p.102

, Hamiltonian cubic graphs H 1 and H 2 with 8 vertices in which there is no community of

=. and .. , , p.106

3. , , p.112

A. , T. , and E. , where the 5 vertices in gray cannot be extended into a community, p.114

.. , A split graph in which there is no partition into two 2-clubs but there is a partition into two cliques in the squared graph. Some edges are dotted in G 2 in order to highlight the partition into two cliques, p.120

, The split graph G defined from the instance I = (X, C) with X = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } and C = {x 1 ?x 2 ?x 3, p.121

.. , The graph G and a set of edges (represented by dotted lines) of minimum size to add to make G having diameter 2 (a minimum dominating set is given in gray), p.123

.. , A split graph and its spanning subtree of diameter 4, p.124

.. , A split graph, a graph almost split of length 1 and a graph almost split of length 2, p.126

.. , where an arrow from a class to another indicates that the first class contains the second one We compare the hardness of Max Independent 2-Clique and Max Independent Set in studied graph classes. Max Independent 2-Clique is NP-hard on graph classes at the top of the figure (hatched area) and is polynomial-time solvable on graph classes at the bottom (non-hatched area) Max Independent Set is NP-hard on graph classes on the left of the figure (dotted area) and is polynomial-time solvable on graph classes on the right (non-dotted area), Relationship among some classes of (connected) graphs, p.131

.. , Two graphs in which S is an independent 2-clique, p.132

, The independent 2-clique S and its (partial) neighborhood selected in the proof of Lemma 7.10. Dotted lines are possible edges so z can be at distance 2 from other vertices in S but those are unimportant for the proof, p.136

, The bipartite graph G , an instance of Max Independent 2-Clique, p.141

.. , The graph G for which the corresponding line graph L(G ) is an instance of Max Independent 2-Clique, p.143

N. Abbas and L. K. Stewart, Clustering bipartite and chordal graphs: Complexity, sequential and parallel algorithms, Discrete Applied Mathematics, vol.91, issue.1-3, pp.1-23, 1999.
DOI : 10.1016/S0166-218X(98)00094-8

J. Abello, M. G. Resende, and S. Sudarsky, Massive Quasi-Clique Detection, Proceedings of the 5th Latin American Symposium on Theoretical Informatics, pp.598-612, 2002.
DOI : 10.1007/3-540-45995-2_51

Y. Ahn, J. P. Bagrow, and S. Lehmann, Link communities reveal multiscale complexity in networks, Nature, vol.80, issue.7307, pp.761-764, 2010.
DOI : 10.1038/msb4100155

R. D. Alba, A graph???theoretic definition of a sociometric clique???, The Journal of Mathematical Sociology, vol.11, issue.1, pp.3-113, 1973.
DOI : 10.1002/bs.3830110409

V. E. Alekseev, On the local restrictions effect on the complexity of finding the graph independence number, Combinatorial-Algebraic Methods in Applied Mathematics, pp.3-13, 1983.

P. Alimonti and V. Kann, Hardness of approximating problems on cubic graphs, Proceedings of the 3rd Italian Conference on Algorithms and Complexity, pp.288-298, 1997.
DOI : 10.1007/3-540-62592-5_80

P. Alimonti and V. Kann, Some APX-completeness results for cubic graphs, Theoretical Computer Science, vol.237, issue.1-2, pp.123-134, 2000.
DOI : 10.1016/S0304-3975(98)00158-3

T. Antal, P. L. Krapivsky, and S. Redner, Dynamics of social balance on networks, Physical Review E, vol.65, issue.3, p.36121, 2005.
DOI : 10.1016/S0378-8733(03)00009-1

Y. Asahiro, K. Iwama, H. Tamaki, and T. Tokuyama, Greedily Finding a Dense Subgraph, Journal of Algorithms, vol.34, issue.2, pp.203-221, 2000.
DOI : 10.1006/jagm.1999.1062

Y. Asahiro, E. Miyano, and K. Samizo, Approximating Maximum Diameter-Bounded Subgraphs, Proceedings of the 9th Latin American Symposium on Theoretical Informatics, pp.615-626, 2010.
DOI : 10.1007/978-3-642-12200-2_53

H. Aziz, F. Brandt, and P. Harrenstein, Fractional hedonic games, Proceedings of the 2014 International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS 2014), pp.5-12, 2014.

B. Balasundaram, S. Butenko, and S. Trukhanov, Novel Approaches for Analyzing Biological Networks, Journal of Combinatorial Optimization, vol.393, issue.1, pp.23-39, 2005.
DOI : 10.1007/978-1-4757-3023-4_1

M. J. Barber, Modularity and community detection in bipartite networks, Physical Review E, vol.2005, issue.6, p.66102, 2007.
DOI : 10.1073/pnas.0605965104

C. Bazgan, J. Chlebíková, C. Dallard, and T. Pontoizeau, Family of graphs without 2-community structure, The 10th International Colloquium on Graph Theory and combinatorics, 2018.

C. Bazgan, J. Chlebíková, and T. Pontoizeau, New Insight into 2-Community Structures in Graphs with Applications in Social Networks, Proceedings of the 9th International Conference on Combinatorial Optimization and Applications, pp.236-250, 2015.
DOI : 10.1016/j.cosrev.2007.05.001
URL : https://hal.archives-ouvertes.fr/hal-01432509

C. Bazgan, J. Chlebíková, and T. Pontoizeau, Structural and Algorithmic Properties of 2-Community Structures, Algorithmica, vol.1, issue.3, pp.1890-1908, 2018.
DOI : 10.1016/j.cosrev.2007.05.001
URL : https://hal.archives-ouvertes.fr/hal-01482319

C. Bazgan, T. Pontoizeau, and Z. Tuza, On the Complexity of Finding a Potential Community, Proceedings of the 10th International Conference on Algorithms and Complexity, pp.80-91, 2017.
DOI : 10.4086/toc.2007.v003a006
URL : https://hal.archives-ouvertes.fr/hal-01512641

C. Bazgan, T. Pontoizeau, and Z. Tuza, Finding a potential community in networks, Theoretical Computer Science

C. Bazgan, Z. Tuza, and D. Vanderpooten, Degree-constrained decompositions of graphs: Bounded treewidth and planarity, Theoretical Computer Science, vol.355, issue.3, pp.389-395, 2006.
DOI : 10.1016/j.tcs.2006.01.024
URL : https://doi.org/10.1016/j.tcs.2006.01.024

C. Bazgan, Z. Tuza, and D. Vanderpooten, The satisfactory partition problem, Discrete Applied Mathematics, vol.154, issue.8, pp.1236-1245, 2006.
DOI : 10.1016/j.dam.2005.10.014
URL : https://hal.archives-ouvertes.fr/hal-00004258

C. Bazgan, Z. Tuza, and D. Vanderpooten, Approximation of satisfactory bisection problems, Journal of Computer and System Sciences, vol.74, issue.5, pp.875-883, 2008.
DOI : 10.1016/j.jcss.2007.12.001
URL : https://doi.org/10.1016/j.jcss.2007.12.001

A. A. Bertossi, Dominating sets for split and bipartite graphs, Information Processing Letters, vol.19, issue.1, pp.37-40, 1984.
DOI : 10.1016/0020-0190(84)90126-1

D. Bilò, L. Gualà, and G. Proietti, Improved approximability and non-approximability results for graph diameter decreasing problems, Theoretical Computer Science, vol.417, pp.12-22, 2012.
DOI : 10.1016/j.tcs.2011.05.014

V. D. Blondel, J. Guillaume, R. Lambiotte, and E. Lefebvre, Fast unfolding of communities in large networks Journal of statistical mechanics: theory and experiment, p.10008, 2008.

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

K. S. Booth and G. S. Lueker, Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms, Journal of Computer and System Sciences, vol.13, issue.3, pp.335-379, 1976.
DOI : 10.1016/S0022-0000(76)80045-1
URL : https://doi.org/10.1016/s0022-0000(76)80045-1

C. Bordenave, M. Lelarge, and L. Massoulié, Non-backtracking Spectrum of Random Graphs: Community Detection and Non-regular Ramanujan Graphs, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pp.1347-1357, 2015.
DOI : 10.1109/FOCS.2015.86
URL : https://hal.archives-ouvertes.fr/hal-01137952

S. P. Borgatti, M. G. Everett, and P. R. Shirey, LS sets, lambda sets and other cohesive subsets, Social Networks, vol.12, issue.4, pp.337-357, 1990.
DOI : 10.1016/0378-8733(90)90014-Z

A. S. Brahim, B. L. Grand, L. Tabourier, and M. Latapy, Citations among blogs in a hierarchy of communities: Method and case study, Journal of Computational Science, vol.2, issue.3, pp.247-252, 2011.
DOI : 10.1016/j.jocs.2011.05.010
URL : https://hal.archives-ouvertes.fr/hal-01146074

U. Brandes, D. Delling, M. Gaertler, R. Görke, M. Hoefer et al., On Finding Graph Clusterings with Maximum Modularity, Proceedings of the 33rd International Conference on Graph-theoretic Concepts in Computer Science, pp.121-132, 2007.
DOI : 10.1007/978-3-540-74839-7_12
URL : http://i11www.ira.uka.de/extra/publications/bdhggnw-fgcmm-07.pdf

S. Cafieri, P. Hansen, and L. Liberti, Improving heuristics for network modularity maximization using an exact algorithm, Discrete Applied Mathematics, vol.163, pp.65-72, 2014.
DOI : 10.1016/j.dam.2012.03.030
URL : https://hal.archives-ouvertes.fr/hal-00935211

A. Cami, H. Balakrishnan, N. Deo, and R. D. Dutton, On the complexity of finding optimal global alliances, Journal of Combinatorial Mathematics and Combinatorial Computing, vol.58, pp.23-31, 2006.

D. Cartwright and F. Harary, Structural balance: a generalization of Heider's theory., Psychological Review, vol.63, issue.5, pp.277-293, 1956.
DOI : 10.1037/h0046049

R. Cazabet, R. Baccour, and M. Latapy, Tracking bitcoin users activity using community detection on a network of weak signals
URL : https://hal.archives-ouvertes.fr/hal-01617992

M. R. Cerioli, L. Faria, T. O. Ferreira, C. A. Martinhon, F. Protti et al., Partition into cliques for cubic graphs: Planar case, complexity and approximation, Discrete Applied Mathematics, vol.156, issue.12, pp.1562270-2278, 2008.
DOI : 10.1016/j.dam.2007.10.015

M. R. Cerioli, L. Faria, T. O. Ferreira, and F. Protti, A note on maximum independent sets and minimum clique partitions in unit disk graphs and penny graphs: complexity and approximation, RAIRO-Theoretical Informatics and Applications, pp.331-346, 2011.
DOI : 10.1109/TC.1981.6312176
URL : http://www.numdam.org/article/ITA_2011__45_3_331_0.pdf

J. Chang, J. Yang, and S. Peng, On the complexity of graph clustering with bounded diameter, 2014 International Computer Science and Engineering Conference (ICSEC), pp.18-22, 2014.
DOI : 10.1109/ICSEC.2014.6978122

M. Chang, L. Hung, C. Lin, and P. Su, Finding large $$k$$ -clubs in undirected graphs, Computing, vol.6, issue.5, pp.739-758, 2013.
DOI : 10.1017/CBO9780511815478

C. Cheng and Y. C. Wei, An improved two-way partitioning algorithm with stable performance (VLSI), IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol.10, issue.12, pp.1502-1511, 2006.
DOI : 10.1109/43.103500

J. Chlebíková, Approximating the maximally balanced connected partition problem in graphs, Information Processing Letters, vol.60, issue.5, pp.225-230, 1996.
DOI : 10.1016/S0020-0190(96)00175-5

T. Choe and C. Park, A k-way Graph Partitioning Algorithm Based on Clustering by Eigenvector, Proceedings of the 4th International Conference of Computational Science, pp.598-601, 2004.
DOI : 10.1007/978-3-540-24687-9_81

S. A. Cook, The complexity of theorem-proving procedures, Proceedings of the third annual ACM symposium on Theory of computing , STOC '71, pp.151-158, 1971.
DOI : 10.1145/800157.805047
URL : http://www.inf.unibz.it/~calvanese/teaching/13-14-tc/material/cook-1971-NP-completeness-of-SAT.pdf

D. G. Corneil, H. Lerchs, and L. S. Burlingham, Complement reducible graphs, Discrete Applied Mathematics, vol.3, issue.3, pp.163-174, 1981.
DOI : 10.1016/0166-218X(81)90013-5
URL : https://doi.org/10.1016/0166-218x(81)90013-5

D. G. Corneil and Y. Perl, Clustering and domination in perfect graphs, Discrete Applied Mathematics, vol.9, issue.1, pp.27-39, 1984.
DOI : 10.1016/0166-218X(84)90088-X
URL : https://doi.org/10.1016/0166-218x(84)90088-x

D. G. Corneil, Y. Perl, and L. Stewart, A Linear Recognition Algorithm for Cographs, SIAM Journal on Computing, vol.14, issue.4, pp.926-934, 1985.
DOI : 10.1137/0214065

B. Courcelle, The monadic second-order logic of graphs III : tree-decompositions, minors and complexity issues, RAIRO -Informatique Théorique et Applications, pp.257-286, 1992.
DOI : 10.1007/BFb0017421

B. Courcelle and J. Engelfriet, Graph structure and monadic second-order logic: a language-theoretic approach, 2012.
DOI : 10.1017/CBO9780511977619
URL : https://hal.archives-ouvertes.fr/hal-00646514

B. Courcelle, J. A. Makowsky, and U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, pp.125-150, 2000.
DOI : 10.1007/s002249910009

B. Courcelle and S. Olariu, Upper bounds to the clique width of graphs, Discrete Applied Mathematics, vol.101, issue.1-3, pp.77-114, 2000.
DOI : 10.1016/S0166-218X(99)00184-5
URL : https://doi.org/10.1016/s0166-218x(99)00184-5

A. Cournier and M. Habib, A new linear algorithm for Modular Decomposition, Proceedings of the 19th International Colloquium on Trees in Algebra and Programming, pp.68-84, 1994.
DOI : 10.1007/BFb0017474

D. Delling, A. Goldberg, T. Pajor, and R. Werneck, Customizable Route Planning, Proceedings of the 10th International Symposium on Experimental Algorithms, pp.376-387, 2011.
DOI : 10.1007/3-540-45643-0_4
URL : http://i11www.ira.uka.de/extra/publications/dw-tdrp-09.pdf

J. S. Deogun, D. Kratsch, and G. Steiner, An approximation algorithm for clustering graphs with dominating diametral path, Information Processing Letters, vol.61, issue.3, pp.121-127, 1997.
DOI : 10.1016/S0020-0190(97)81663-8

R. G. Downey and M. R. Fellows, Fixed-parameter tractability and completeness II: On completeness for W[1], Theoretical Computer Science, vol.141, issue.1-2, pp.109-131, 1995.
DOI : 10.1016/0304-3975(94)00097-3
URL : https://doi.org/10.1016/0304-3975(94)00097-3

R. G. Downey and M. R. Fellows, Parameterized complexity, 2012.
DOI : 10.1007/978-1-4612-0515-9

J. Duch and A. Arenas, Community detection in complex networks using extremal optimization, Physical Review E, vol.2004, issue.2, p.27104, 2005.
DOI : 10.1038/nature03288
URL : http://arxiv.org/pdf/cond-mat/0501368

A. Dumitrescu and J. Pach, Minimum Clique Partition in Unit Disk Graphs, Graphs and Combinatorics, pp.399-411, 2011.
DOI : 10.4086/toc.2007.v003a006
URL : http://arxiv.org/pdf/0909.1552

D. Easley and J. Kleinberg, Networks, Crowds, and Markets: Reasoning About a Highly Connected World, 2010.
DOI : 10.1017/CBO9780511761942

V. Estivill-castro and M. Parsa, On connected two communities, Proceedings of the 36th Australasian Computer Science Conference, pp.23-30, 2013.

V. Estivill-castro and M. Parsa, Hardness and tractability of detecting connected communities, Proceedings of the Australasian Computer Science Week Multiconference on, ACSW '16, pp.1-25, 2016.
DOI : 10.1145/1556460.1556466

T. S. Evans and R. Lambiotte, Line graphs, link partitions, and overlapping communities, Physical Review E, vol.8, issue.1, p.16105, 2009.
DOI : 10.1088/1367-2630/10/5/053039
URL : http://spiral.imperial.ac.uk/bitstream/10044/1/9026/1/link090710.pdf

M. G. Everett and S. Borgatti, Role colouring a graph, Mathematical Social Sciences, vol.21, issue.2, pp.183-188, 1991.
DOI : 10.1016/0165-4896(91)90080-B

B. S. Everitt, S. Landau, M. Leese, and D. Stahl, Cluster analysis, 2011.
DOI : 10.1002/9780470977811
URL : https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470977811.fmatter

U. Feige, D. Peleg, and G. Kortsarz, The Dense k -Subgraph Problem, Algorithmica, vol.29, issue.3, pp.410-421, 2001.
DOI : 10.1007/s004530010050
URL : http://www.openu.ac.il/Personal_sites/download/accepted.ps

A. Feldmann and L. Foschini, Balanced Partitions of Trees and Applications, Algorithmica, vol.15, issue.11, pp.354-376, 2015.
DOI : 10.1109/34.244673
URL : https://hal.archives-ouvertes.fr/hal-00678179

H. Fernau and D. Raible, Alliances in graphs: a complexity-theoretic study, Proceedings on the 33rd International Conference on Current Trends in Theory and Practice of Computer Science, pp.61-70, 2007.

H. Fernau and J. A. Rodríguez-velázquez, A survey on alliances and related parameters in graphs, Electronic Journal of Graph Theory and Applications, vol.2, issue.1, pp.70-86, 2014.
DOI : 10.5614/ejgta.2014.2.1.7
URL : https://doi.org/10.5614/ejgta.2014.2.1.7

J. Fiala and D. Paulusma, A complete complexity classification of the role assignment problem, Theoretical Computer Science, vol.349, issue.1, pp.67-81, 2005.
DOI : 10.1016/j.tcs.2005.09.029
URL : https://doi.org/10.1016/j.tcs.2005.09.029

G. Flake, S. Lawrence, and C. Giles, Efficient identification of Web communities, Proceedings of the sixth ACM SIGKDD international conference on Knowledge discovery and data mining , KDD '00, pp.150-160, 2000.
DOI : 10.1145/347090.347121
URL : http://gdit.iiit.net/wdkm/kdd2000.ps

H. Fleischner, E. Mujuni, D. Paulusma, and S. Szeider, Covering graphs with few complete bipartite subgraphs, Theoretical Computer Science, vol.410, pp.21-232045, 2009.
DOI : 10.1007/978-3-540-77050-3_28

L. R. Ford-jr and D. R. Fulkerson, Maximal flow through a network, Canadian Journal of Mathematics, pp.399-404, 1956.

S. Fortunato, Community detection in graphs, Physics Reports, vol.486, issue.3-5, pp.3-575, 2010.
DOI : 10.1016/j.physrep.2009.11.002
URL : http://arxiv.org/pdf/0906.0612v1.pdf

S. Fortunato and D. Hric, Community detection in networks: A user guide, Physics Reports, vol.659, pp.1-44, 2016.
DOI : 10.1016/j.physrep.2016.09.002
URL : http://arxiv.org/pdf/1608.00163

A. Frank, Some polynomial algorithms for certain graphs and hypergraphs, Congressus Numerantium No. XV, pp.3-13, 1976.

L. C. Freeman, A Set of Measures of Centrality Based on Betweenness, Sociometry, vol.40, issue.1, pp.35-41, 1977.
DOI : 10.2307/3033543

T. Gallai, Transitiv orientierbare Graphen, Acta Mathematica Academiae Scientiarum Hungaricae, vol.51, issue.1-2, pp.25-66, 1967.
DOI : 10.4153/CJM-1964-055-5

Y. Gao, D. Hare, and J. Nastos, The parametric complexity of graph diameter augmentation, Discrete Applied Mathematics, vol.161, issue.10-11, pp.10-111626, 2013.
DOI : 10.1016/j.dam.2013.01.016
URL : https://doi.org/10.1016/j.dam.2013.01.016

M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, 1979.

M. R. Garey, D. S. Johnson, and L. Stockmeyer, Some simplified NP-complete graph problems, Theoretical Computer Science, vol.1, issue.3, pp.237-267, 1976.
DOI : 10.1016/0304-3975(76)90059-1
URL : https://doi.org/10.1016/0304-3975(76)90059-1

B. Gaume, Balades aléatoires dans les petits mondes lexicaux, I3 Information Interaction Intelligence, pp.39-96, 2004.

M. Gerber and D. Kobler, Partitioning a graph to satisfy all vertices, 1998.

M. Gerber and D. Kobler, Algorithmic approach to the satisfactory graph partitioning problem, European Journal of Operational Research, vol.125, issue.2, pp.283-291, 2000.
DOI : 10.1016/S0377-2217(99)00459-2

M. Girvan and M. E. Newman, Community structure in social and biological networks, Proceedings of the National Academy of Sciences, pp.7821-7826, 2002.
DOI : 10.1086/285382
URL : http://www.pnas.org/content/99/12/7821.full.pdf

A. V. Goldberg, Finding a maximum density subgraph, 1984.

O. Goldschmidt and D. S. Hochbaum, Polynomial algorithm for the k-cut problem, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science, pp.444-451, 1988.
DOI : 10.1109/SFCS.1988.21960

P. Golovach, P. Heggernes, D. Kratsch, and A. Rafiey, Finding clubs in graph classes, Discrete Applied Mathematics, vol.174, pp.57-65, 2014.
DOI : 10.1016/j.dam.2014.04.016

M. González, H. Herrmann, J. Kertész, and T. Vicsek, Community structure and ethnic preferences in school friendship networks. Physica A: Statistical Mechanics and its Applications, pp.307-316, 2007.

M. Grötschel, L. Lovász, and A. Schrijver, Stable sets in graphs, Geometric Algorithms and Combinatorial Optimization, pp.272-303, 1988.

L. Gulikers, M. Lelarge, and L. Massoulié, Abstract, Advances in Applied Probability, vol.49, issue.03, pp.49686-721, 2017.
DOI : 10.1214/14-AOS1274

U. I. Gupta, D. T. Lee, and J. Y. Leung, Efficient algorithms for interval graphs and circular-arc graphs, Networks, vol.9, issue.4, pp.459-467, 1982.
DOI : 10.1016/S0022-0000(76)80045-1

M. Habib and C. Paul, A simple linear time algorithm for cograph recognition, Discrete Applied Mathematics, vol.145, issue.2, pp.183-197, 2005.
DOI : 10.1016/j.dam.2004.01.011
URL : https://hal.archives-ouvertes.fr/lirmm-00105298

M. Habib and C. Paul, A survey of the algorithmic aspects of modular decomposition, Computer Science Review, vol.4, issue.1, pp.41-59, 2010.
DOI : 10.1016/j.cosrev.2010.01.001

F. Harary, On the notion of balance of a signed graph., The Michigan Mathematical Journal, vol.2, issue.2, pp.143-146, 1953.
DOI : 10.1307/mmj/1028989917

S. Hartung, C. Komusiewicz, and A. Nichterlein, Parameterized algorithmics and computational experiments for finding 2-clubs, Proceedings of the 7th International Symposium on Parameterized and Exact Computation, pp.231-241, 2012.
DOI : 10.7155/jgaa.00352
URL : http://fpt.akt.tu-berlin.de/publications/sclubImplementation.pdf

S. Hartung, C. Komusiewicz, A. Nichterlein, and O. Suchý, On structural parameterizations for the 2-club problem, Discrete Applied Mathematics, vol.185, issue.C, pp.79-92, 2015.
DOI : 10.1016/j.dam.2014.11.026
URL : http://fpt.akt.tu-berlin.de/publications/sclubHardness.pdf

J. He, J. Hopcroft, H. Liang, S. Suwajanakorn, and L. Wang, Detecting the Structure of Social Networks Using (??,??)-Communities, Proceedings of the 8th International Workshop on Algorithms and Models for the Web Graph, pp.26-37, 2011.
DOI : 10.1016/j.cosrev.2007.05.001

P. Heggernes and D. Kratsch, Linear-time certifying recognition algorithms and forbidden induced subgraphs, Nordic Journal of Computing, vol.14, pp.87-108, 2007.

F. Heider, Attitudes and Cognitive Organization, The Journal of Psychology, vol.21, issue.1, pp.107-112, 1946.
DOI : 10.1037/h0055425

B. Hendrickson and R. W. Leland, A multi-level algorithm for partitioning graphs, Proceedings of the 1995 ACM/IEEE Conference on Supercomputing, pp.28-41, 1995.

S. D. Hochbaum, Approximation Algorithms for NP-hard Problems, 1997.

L. H. Jamieson, S. T. Hedetniemi, and A. A. Mcrae, The algorithmic complexity of alliances in graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, vol.68, pp.137-150, 2009.

S. Jespersen, I. M. Sokolov, and A. Blumen, Relaxation properties of small-world networks, Physical Review E, vol.43, issue.3, pp.4405-4408, 2000.
DOI : 10.1051/jphyslet:019820043017062500
URL : http://arxiv.org/pdf/cond-mat/0004214v1.pdf

P. F. Jonsson and P. A. Bates, Global topological features of cancer proteins in the human interactome, Bioinformatics, vol.5, issue.18, pp.2291-2297, 2006.
DOI : 10.1186/1471-2148-5-24

P. F. Jonsson, T. Cavanna, D. Zicha, and P. A. Bates, Cluster analysis of networks generated through homology: automatic identification of important protein communities involved in cancer metastasis, BMC Bioinformatics, vol.7, issue.1, 2006.

R. M. Karp, Reducibility among Combinatorial Problems, pp.85-103, 1972.
DOI : 10.1007/978-3-540-68279-0_8

G. Karypis and V. Kumar, Parallel multilevel k-way partitioning scheme for irregular graphs, Proceedings of the 1996 ACM/IEEE conference on Supercomputing (CDROM) , Supercomputing '96, pp.96-129, 1998.
DOI : 10.1145/369028.369103

A. Kenneth and H. Wolfgang, Every planar map is four colorable : Part i. discharging, Illinois Journal of Mathematics, vol.21, pp.429-490, 1977.

B. W. Kernighan and S. Lin, An Efficient Heuristic Procedure for Partitioning Graphs, Bell System Technical Journal, vol.49, issue.2, pp.291-307, 1970.
DOI : 10.1002/j.1538-7305.1970.tb01770.x

S. Khanna, R. Motwani, M. Sudan, and U. V. Vazirani, On Syntactic versus Computational Views of Approximability, SIAM Journal on Computing, vol.28, issue.1, pp.164-191, 1998.
DOI : 10.1137/S0097539795286612

J. Kleinberg, An impossibility theorem for clustering, Proceedings of the 15th International Conference on Neural Information Processing Systems, pp.463-470, 2002.

C. Komusiewicz, Multivariate Algorithmics for Finding Cohesive Subnetworks, Algorithms, vol.48, issue.1, p.21, 2016.
DOI : 10.1145/1597036.1597044

P. Kristiansen, S. Hedetniemi, and S. Hedetniemi, Alliances in graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, vol.48, pp.157-177, 2004.

C. Kuratowski, Sur le probl??me des courbes gauches en Topologie, Fundamenta Mathematicae, vol.15, issue.1, pp.271-283, 1930.
DOI : 10.4064/fm-15-1-271-283
URL : https://www.impan.pl/shop/publication/transaction/download/product/92829?download.pdf

S. Laan, M. Marx, and R. J. Mokken, Close communities in social networks: boroughs and 2-clubs. Social Network Analysis and Mining, p.20, 2016.
DOI : 10.2139/ssrn.2686127
URL : https://pure.uva.nl/ws/files/2741822/173165_Laan_e_a_2016_Close_Communities_SNAM.pdf

A. Lancichinetti and S. Fortunato, Community detection algorithms: A comparative analysis, Physical Review E, vol.6, issue.5, p.56117, 2009.
DOI : 10.1103/PhysRevLett.96.114102

E. Lawler, Cutsets and partitions of hypergraphs, Networks, vol.19, issue.3, pp.275-285, 1973.
DOI : 10.1145/321694.321699

J. Leskovec, K. J. Lang, A. Dasgupta, and M. W. Mahoney, Statistical properties of community structure in large social and information networks, Proceeding of the 17th international conference on World Wide Web , WWW '08, pp.695-704, 2008.
DOI : 10.1145/1367497.1367591

J. Leskovec, K. J. Lang, and M. Mahoney, Empirical comparison of algorithms for network community detection, Proceedings of the 19th international conference on World wide web, WWW '10, pp.631-640, 2010.
DOI : 10.1145/1772690.1772755
URL : http://cs.stanford.edu/%7Ejure/pubs/communities-www10.pdf

C. Li, S. Mccormick, and D. Simchi-levi, On the minimum-cardinality-bounded-diameter and the bounded-cardinality-minimum-diameter edge addition problems, Operations Research Letters, vol.11, issue.5, pp.303-308, 1992.
DOI : 10.1016/0167-6377(92)90007-P

D. Lokshtanov, N. Misra, G. Philip, M. S. Ramanujan, and S. Saurabh, Hardness of r-dominating set on Graphs of Diameter (r???+???1), Proceedings of the 8th International Symposium on Parameterized and Exact Computation, pp.255-267, 2013.
DOI : 10.1007/978-3-319-03898-8_22
URL : http://www.ii.uib.no/~daniello/papers/rDomSetHard.pdf

L. Lovász, A characterization of perfect graphs, Journal of Combinatorial Theory, Series B, vol.13, issue.2, pp.95-98, 1972.
DOI : 10.1016/0095-8956(72)90045-7

F. Luccio and M. Sami, On the Decomposition of Networks in Minimally Interconnected Subnetworks, IEEE Transactions on Circuit Theory, vol.16, issue.2, pp.184-188, 1969.
DOI : 10.1109/TCT.1969.1082924

S. Mancoridis, B. S. Mitchell, C. Rorres, Y. Chen, and E. R. Gansner, Using automatic clustering to produce high-level system organizations of source code, Proceedings. 6th International Workshop on Program Comprehension. IWPC'98 (Cat. No.98TB100242), pp.45-52, 1998.
DOI : 10.1109/WPC.1998.693283
URL : http://www.cs.drexel.edu/~smancori/research/papers/iwpc98.pdf

L. Massoulié, Community detection thresholds and the weak Ramanujan property, Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC '14, pp.694-703, 2014.
DOI : 10.1017/S0963548309990514

H. Matsuda, T. Ishihara, and A. Hashimoto, Classifying molecular sequences using a linkage graph with their pairwise similarities, Theoretical Computer Science, vol.210, issue.2, pp.305-325, 1999.
DOI : 10.1016/S0304-3975(98)00091-7
URL : https://doi.org/10.1016/s0304-3975(98)00091-7

N. Mishra, R. Schreiber, I. Stanton, and R. E. Tarjan, Finding Strongly Knit Clusters in Social Networks, Internet Mathematics, vol.5, issue.1-2, pp.155-174, 2008.
DOI : 10.1080/15427951.2008.10129299
URL : http://www.internetmathematicsjournal.com/article/1452.pdf

R. J. Mokken, Cliques, clubs and clans, Quality & Quantity, vol.15, issue.2, pp.161-173, 1979.
DOI : 10.1007/BF00139635

R. J. Mokken, E. M. Heemskerk, and S. Laan, Close communication and 2-clubs in corporate networks: Europe 2010, Social Network Analysis and Mining, vol.18, issue.1, p.40, 2016.
DOI : 10.1017/CBO9780511815478
URL : https://pure.uva.nl/ws/files/2699803/175441_Mokken_e_a_2016_CloseComm_Corp_Europe2010_SNAM.pdf

J. W. Moon and L. Moser, On cliques in graphs, Israel Journal of Mathematics, vol.3, issue.1, pp.23-28, 1965.
DOI : 10.1007/BF02760024

M. Newman, Detecting community structure in networks, The European Physical Journal B - Condensed Matter, vol.38, issue.2, pp.321-330, 2004.
DOI : 10.1140/epjb/e2004-00124-y

M. E. Newman, Mixing patterns in networks, Physical Review E, vol.42, issue.2, p.26126, 2003.
DOI : 10.1137/S0036144500371907
URL : http://arxiv.org/pdf/cond-mat/0209450

M. E. Newman and M. Girvan, Finding and evaluating community structure in networks, Physical Review E, vol.65, issue.2, p.26113, 2004.
DOI : 10.1103/PhysRevE.68.065103
URL : http://arxiv.org/pdf/cond-mat/0308217

R. Niedermeier, Invitation to fixed-parameter algorithms, 2006.
DOI : 10.1093/acprof:oso/9780198566076.001.0001

M. Olsen, A general view on computing communities, Mathematical Social Sciences, vol.66, issue.3, pp.331-336, 2013.
DOI : 10.1016/j.mathsocsci.2013.07.002

F. M. Pajouh and B. Balasundaram, On inclusionwise maximal and maximum cardinality <mml:math altimg="si74.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>k</mml:mi></mml:math>-clubs in graphs, Discrete Optimization, vol.9, issue.2, pp.84-97, 2012.
DOI : 10.1016/j.disopt.2012.02.002

G. Palla, A. Barabási, and T. Vicsek, Quantifying social group evolution, Nature, vol.21, issue.7136, p.446, 2007.
DOI : 10.1038/nature05670
URL : http://arxiv.org/pdf/0704.0744

G. Palla, I. Derényi, I. Farkas, and T. Vicsek, Uncovering the overlapping community structure of complex networks in nature and society, Nature, vol.433, issue.7043, pp.814-818, 2005.
DOI : 10.1038/nature03248
URL : http://arxiv.org/pdf/physics/0506133

C. H. Papadimitriou and M. Yannakakis, Optimization, approximation, and complexity classes, Journal of Computer and System Sciences, vol.43, issue.3, pp.425-440, 1991.
DOI : 10.1016/0022-0000(91)90023-X

A. Parley, S. Hedetniemi, and A. Proskurowski, Partitioning trees: Matching, domination, and maximum diameter, International Journal of Computer & Information Sciences, vol.6, issue.1, pp.55-61, 1981.
DOI : 10.1007/BF00978378

I. A. Pirwani and M. R. Salavatipour, A weakly robust PTAS for minimum clique partition in unit disk graphs, Algorithmica, vol.62, pp.3-41050, 2012.
DOI : 10.1007/978-3-642-13731-0_19
URL : http://arxiv.org/pdf/0904.2203.pdf

J. Plesnik, The complexity of designing a network with minimum diameter, Networks, vol.23, issue.1, pp.77-85, 1981.
DOI : 10.1145/321958.321975

P. Pons and M. Latapy, Computing Communities in Large Networks Using Random Walks, pp.284-293, 2005.
DOI : 10.1007/11569596_31
URL : http://arxiv.org/pdf/cond-mat/0412368

A. Puig-centelles, O. Ripolles, and M. Chover, Surveying the identification of communities, International Journal of Web Based Communities, vol.4, issue.3, pp.334-347, 2008.
DOI : 10.1504/IJWBC.2008.019193

F. Radicchi, C. Castellano, F. Cecconi, V. Loreto, and D. Parisi, Defining and identifying communities in networks, Proceedings of the National Academy of Sciences, vol.68, issue.4, pp.2658-2663, 2004.
DOI : 10.1080/0022250X.2001.9990249
URL : http://www.pnas.org/content/101/9/2658.full.pdf

V. Raman and S. Saurabh, Triangles, 4-Cycles and Parameterized (In-)Tractability, Proceedings of the 10th Scandinavian Workshop on Algorithm Theory, pp.304-315, 2006.
DOI : 10.1007/11785293_29

F. S. Roberts and L. Sheng, How hard is it to determine if a graph has a 2-role assignment? Networks, pp.67-73, 2001.
DOI : 10.1002/1097-0037(200103)37:2<67::aid-net1>3.0.co;2-9
URL : http://dimacs.rutgers.edu/People/Staff/froberts/135.ps

S. E. Schaeffer, Graph clustering, Computer Science Review, vol.1, issue.1, pp.27-64, 2007.
DOI : 10.1016/j.cosrev.2007.05.001

A. Schäfer, C. Komusiewicz, H. Moser, and R. Niedermeier, Parameterized computational complexity of finding small-diameter subgraphs. Optimization Letters, pp.883-891, 2012.

A. Schoone, H. Bodlaender, and J. Leeuwen, Diameter increase caused by edge deletion, Journal of Graph Theory, vol.11, issue.3, pp.409-427, 1987.
DOI : 10.1090/psapm/034/846852
URL : https://dspace.library.uu.nl/bitstream/1874/16289/2/schoone_85_diameter_increase.pdf

S. B. Seidman, LS sets as cohesive subsets of graphs and hypergraphs, Mathematical Social Sciences, vol.6, issue.1, pp.87-91, 1983.
DOI : 10.1016/0165-4896(83)90048-3

S. B. Seidman, Network structure and minimum degree, Social Networks, vol.5, issue.3, pp.269-287, 1983.
DOI : 10.1016/0378-8733(83)90028-X

S. B. Seidman and B. L. Foster, A graph???theoretic generalization of the clique concept*, The Journal of Mathematical Sociology, vol.33, issue.1, pp.139-154, 1978.
DOI : 10.1002/bs.3830110409

D. Shmoys, Cut problems and their application to divide-and-conquer, Approximation Algorithms for NP-hard Problems, pp.192-235, 1997.

J. Sigarreta, S. Bermudo, and H. Fernau, On the complement graph and defensive <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>k</mml:mi></mml:math>-alliances, Discrete Applied Mathematics, vol.157, issue.8, pp.1687-1695, 2009.
DOI : 10.1016/j.dam.2008.12.006

S. Tsukiyama, M. Ide, H. Ariyoshi, and I. Shirakawa, A New Algorithm for Generating All the Maximal Independent Sets, SIAM Journal on Computing, vol.6, issue.3, pp.505-517, 1977.
DOI : 10.1137/0206036

V. V. Vazirani, Approximation Algorithms, 2001.
DOI : 10.1007/978-3-662-04565-7

L. Wang, J. Hopcroft, J. He, H. Liang, and S. Suwajanakorn, Extracting the Core Structure of Social Networks Using (??, ??)-Communities, Internet Mathematics, vol.9, issue.1, pp.58-81, 2013.
DOI : 10.1080/15427951.2012.678187
URL : http://www.internetmathematicsjournal.com/article/1533.pdf

J. Xie, S. Kelley, and B. K. Szymanski, Overlapping community detection in networks, ACM Computing Surveys, vol.45, issue.4, pp.1-4335, 2013.
DOI : 10.1145/2501654.2501657
URL : http://arxiv.org/pdf/1110.5813

J. Yang and J. Leskovec, Defining and evaluating network communities based on ground-truth, Proceedings of the ACM SIGKDD Workshop on Mining Data Semantics , MDS '12, pp.1-3, 2012.
DOI : 10.1109/icdm.2012.138

W. W. Zachary, An Information Flow Model for Conflict and Fission in Small Groups, Journal of Anthropological Research, vol.33, issue.4, pp.452-473, 1977.
DOI : 10.1086/jar.33.4.3629752
URL : http://arxiv.org/pdf/1707.03587

H. Zhou and R. Lipowsky, Network Brownian Motion: A New Method to Measure Vertex-Vertex Proximity and to Identify Communities and Subcommunities, Proceedings of the 4th International Conference on Computational Science, pp.1062-1069, 2004.
DOI : 10.1007/978-3-540-24688-6_137
URL : https://link.springer.com/content/pdf/10.1007%2F978-3-540-24688-6_137.pdf

D. Zuckerman, Linear degree extractors and the inapproximability of max clique and chromatic number, Proceedings of the thirty-eighth annual ACM symposium on Theory of computing , STOC '06, pp.103-128, 2007.
DOI : 10.1145/1132516.1132612
URL : http://theoryofcomputing.org/articles/main/v003/a006/a006.pdf