Graphs Without Large Apples and the Maximum Weight Independent Set Problem
 Vadim V. Lozin,
 Martin Milanič,
 Christopher Purcell
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
An apple A _{ k } is the graph obtained from a chordless cycle C _{ k } of length k ≥ 4 by adding a vertex that has exactly one neighbor on the cycle. The class of applefree graphs is a common generalization of clawfree graphs and chordal graphs, two classes enjoying many attractive properties, including polynomialtime solvability of the maximum weight independent set problem. Recently, Brandstädt et al. showed that this property extends to the class of applefree graphs. In the present paper, we study further generalization of this class called graphs without large apples: these are (A _{ k }, A _{ k+1}, . . .)free graphs for values of k strictly greater than 4. The complexity of the maximum weight independent set problem is unknown even for k = 5. By exploring the structure of graphs without large apples, we discover a sufficient condition for clawfreeness of such graphs. We show that the condition is satisfied by boundeddegree and apexminorfree graphs of sufficiently large treewidth. This implies an efficient solution to the maximum weight independent set problem for those graphs without large apples, which either have bounded vertex degree or exclude a fixed apex graph as a minor.
 Alekseev, V.E.: On the number of maximal independent sets in graphs from hereditary classes, Combinatorialalgebraic methods in discrete optimization, University of Nizhny Novgorod, pp. 5–8 (in Russian) (1991)
 Alekseev, V.E. (2004) A polynomial algorithm for finding the largest independent sets in forkfree graphs. Discret. Appl. Math. 135: pp. 316 CrossRef
 Alekseev, V.E., Lozin, Malyshev, V.D., Milanič, M.: The maximum independent set problem in planar graphs, In: Proceedings of 33rd International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, vol. 5162, pp. 96–107 (2008)
 Arnborg, S., Proskurowski, A. (1989) Linear time algorithms for NPhard problems restricted to partial ktrees. Discret. Appl. Math. 23: pp. 1124 CrossRef
 Bodlaender, H.L. (1998) A partial karboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209: pp. 145 CrossRef
 Bodlaender, H.L., Thilikos, D.M. (1997) Treewidth for graphs with small chordality. Discret. Appl. Math. 79: pp. 4561 CrossRef
 Brandstädt, A., Hoà àng, C. (2007) On clique separators, nearly chordal graphs, and the maximum weight stable set problem. Theor. Comput. Sci. 389: pp. 295306 CrossRef
 Brandstädt, A., Lozin, V., Mosca, R. (2010) Independent sets of maximum weight in applefree graphs. SIAM J. Discret. Math. 24: pp. 239254 CrossRef
 Chudnovsky, M., Seymour, P. (2007) Clawfree graphs I—orientable prismatic graphs. J. Combin. Theory Ser. B 97: pp. 867901 CrossRef
 Chudnovsky, M., Seymour, P. (2008) Clawfree graphs II—nonorientable prismatic graphs. J. Combin. Theory Ser. B 98: pp. 249290 CrossRef
 Chudnovsky, M., Seymour, P. (2008) Clawfree graphs III—circular interval graphs. J. Combin. Theory Ser. B 98: pp. 812834 CrossRef
 Chudnovsky, M., Seymour, P. (2008) Clawfree graphs IV—decomposition theorem. J. Combin. Theory Ser. B. 98: pp. 839938 CrossRef
 Chudnovsky, M., Seymour, P. (2008) Clawfree graphs V—global structure. J. Combin. Theor. Ser. B 98: pp. 13731410 CrossRef
 Demaine, E.D., Hajiaghayi, M.T. (2004) Diameter and treewidth in minorclosed graph families, revisited. Algorithmica 40: pp. 211215 CrossRef
 Edmonds, J. (1965) Maximum matching and a polyhedron with 0,1vertices. J. Res. Nat. Bur. Standards Sect. B 69: pp. 125130 CrossRef
 Faenza, Y., Oriolo, G., Stauffer, G.: An algorithmic decomposition of clawfree graphs leading to an O(n ^{3})algorithm for the weighted stable set problem. In: Proceedings of TwentySecond Annual ACMSIAM Symposium on Discrete Algorithms SODA’, vol. 11, pp. 630–646 (2011)
 Farber, M., Hujter, M., Tuza, Z. (1993) An upper bound on the number of cliques in a graph. Networks 23: pp. 207210 CrossRef
 Frank, A. (1976) Some polynomial algorithms for certain graphs and hypergraphs. Congr. Numer. XV: pp. 211226
 Gavril, F. (1972) Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Comput. 1: pp. 180187 CrossRef
 Gavril, F. (1974) The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Combin. Theor. Ser. B 16: pp. 4756 CrossRef
 Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graphs. Topics on perfect graphs, NorthHolland Mathematical Studies, vol. 88, pp. 325–356. NorthHolland, Amsterdam (1984)
 Hertz, A., Werra, D. (1993) On the stability number of AHfree graphs. J. Graph. Theor. 17: pp. 5363 CrossRef
 Kamiński, M., Lozin, V.V., Milanič, M. (2009) Recent developments on graphs of bounded cliquewidth. Discret. Appl. Math. 157: pp. 27472761 CrossRef
 Lovász, L., Plummer, M.D. (1986) Matching theory. Ann. Discret. Math. 29: pp. 295352
 Lozin, V., Milanič, M. (2008) A polynomial algorithm to find an independent set of maximum weight in a forkfree graph. J. Discret. Algorithm 6: pp. 595604 CrossRef
 Lozin, V., Milanič, M.: Maximum independent sets in graphs of low degree. In: Proceedings of Eighteenth Annual ACMSIAM Symposium on Discrete Algorithms SODA’, vol. 07, pp. 874–880 (2007)
 Lozin, V.V., Mosca, R. (2012) Maximum regular induced subgraphs in 2P 3free graphs. Theor. Comput. Sci. 460: pp. 2633 CrossRef
 Minty, G.J. (1980) On maximal independent sets of vertices in clawfree graphs. J. Combin. Theory Ser. B 28: pp. 284304 CrossRef
 Nakamura, D., Tamura, A. (2001) A revision of Minty’s algorithm for finding a maximum weight stable set of a clawfree graph. J. Oper. Res. Soc. Japan 44: pp. 194204
 Oriolo, G., Pietropaoli, U., Stauffer, G.: A new algorithm for the maximum weighted stable set problem in clawfree graphs. In: Proceedings of IPCO 2008, Bertinoro. Lecture Notes in Computer Science. vol. 5035, pp. 96–107 (2008)
 Robertson, N., Seymour, P.D. (1986) Graph minors. V. Excluding a planar graph. J. Comb. Theor. B 41: pp. 92114 CrossRef
 Robertson, N., Seymour, P.D. (1993) Graph searching and a minmax theorem for treewidth. J. Comb. Theory B 58: pp. 2233 CrossRef
 Tarjan, R.E. (1985) Decomposition by clique separators. Discret. Math. 55: pp. 221232 CrossRef
 Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I. (1977) A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6: pp. 505517 CrossRef
 Whitesides, S.H. (1981) An algorithm for finding clique cutsets. Inform. Process. Lett. 12: pp. 3132 CrossRef
 Title
 Graphs Without Large Apples and the Maximum Weight Independent Set Problem
 Journal

Graphs and Combinatorics
Volume 30, Issue 2 , pp 395410
 Cover Date
 20140301
 DOI
 10.1007/s003730121263y
 Print ISSN
 09110119
 Online ISSN
 14355914
 Publisher
 Springer Japan
 Additional Links
 Topics
 Keywords

 Clawfree graphs
 Chordal graphs
 Independent set
 polynomial algorithm
 Industry Sectors
 Authors

 Vadim V. Lozin ^{(1)}
 Martin Milanič ^{(2)} ^{(3)}
 Christopher Purcell ^{(1)}
 Author Affiliations

 1. DIMAP and Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
 2. University of Primorska, UP IAM, Muzejski trg 2, SI6000, Koper, Slovenia
 3. University of Primorska, UP FAMNIT, Glagoljaška 8, SI6000, Koper, Slovenia