Algorithms for the Recognition of Net-free Graphs and
for Computing Maximum Cardinality Matchings in
Claw-free Graphs
Mihai TĂLMACIU1, Victor LEPIN2
1“Vasile Alecsandri” University of Bacău,
Bacău, Romania
mtalmaciu@ub.ro
2 The Institute of Mathematics of the National Academy of Sciences of Belarus
lepin@im.bas-net.by
Abstract: During the last decades, numerous studies have been undertaken on the classes of net-free graphs, claw-free graphs, and the relationship between them. The notion of weakly decomposition (a partition of the set of vertices in three classes A, B, C such that A induces a connected graph and C is totally adjacent to B and totally non-adjacent to A) and the study of its properties allow us to obtain several important results such as: characterization of cographs, {P4,C4}-free and paw-free graphs. In this article, we give a characterization of net-free graphs, a characterization of claw-free graphs, using weakly decomposition. Also, we give a recognition algorithm for net-free graphs, an algorithm for determining a maximum matching in claw-free graphs, comparable with existing algorithms in terms of complexity, but using weakly decomposition.
Keywords: net-free graphs, claw-free graphs, asteroidal triple-free graphs, weakly decomposition, recognition algorithm, maximum matching in graphs.
>>Full text
CITE THIS PAPER AS:
Mihai TĂLMACIU, Victor LEPIN, Algorithms for the Recognition of Net-free Graphs and for Computing Maximum Cardinality Matchings in Claw-free Graphs, Studies in Informatics and Control, ISSN 1220-1766, vol. 23 (2), pp. 183-188, 2014. https://doi.org/10.24846/v23i2y201406
- Introduction
A graph is claw-free if it has no induced subgraph isomorphic to the claw, i.e., the four-vertex star .
A net is a graph obtained from a triangle by attaching to each vertex a new dangling edge.
The interval graphs [24], permutation graphs [16] and co-comparability graphs [18] have a linear structure. Each of these classes is a subfamily of the asteroidal triple graphs (AT-free graphs, for short). An independent set of three vertices is called an asteroidal triple if between any pair in the triple there exists a path that avoids the neighbourhood of the third. AT-free graphs were introduced by Lekkerkerker and Boland [24]. Corneil, Olariu and Stewart showed a number of results on the linear structure of AT-free [8, 9, 10].
A maximal subclass of a class of net-free graphs is the class (claw,net)-free graphs (CN-free graphs, for short). Also note that CN-free graphs are exactly the Hamiltonian-hereditary graphs[13] (was cited in [4]). CN-free graphs turn out to be closely related to AT-free graphs form their structure properties [4]. There are, however, few results about the structure of these graphs [4]. In [4] the authors give results on the linear and circular structure of CN-free graphs. AT-free graphs can be generalized in a manner obvious to admit circular structure [4]. CN-free graphs were introduced by Duffus [14]. Although CN-free graphs seems to be quite restrictive, it contains a couple of families of graphs that are interesting in their own right.
In this paper we give an algorithm for the recognition of net-free graph of complexity O(n(n+ m1,63)). Also, we give an interesting property of claw-free graph that leads to a algorithm for the construction a maximum matching in claw-free graphs.
The content of the paper is organized as follows. In Preliminaries, we give the usual terminology in graph theory. In Section 3 we give a characterization of net-free graphs and a recognition algorithm using the weakly decomposition. In Section 4 we determine a maximum matching in the claw-free graph. Ideas for future work conclude the paper.
REFERENCES
- ALON, N., R. YUSTER, U. ZWICK, Finding and Counting Given Length Cycles, Proc. 2nd European Symposium on Algorithms, Utrecht, The Netherlands, 1994, pp. 354-364
- ASRATIAN, A. S., Every 3-connected, Locally Connected, Claw-free Graph is Hamilton-connected, J. Graph Theory, vol. 23, 1996, pp. 191-201.
- BERGE, C. Graphs, Amsterdam, 1985.
- BRANDSTEDTA, A., F. F. DRĂGAN, On Linear and Circular Structure of (Claw, Net)-free Graphs, Discrete Applied Mathematics, vol. 129(23), 2003, pp. 285-303.
- BRANDSTADT, A., F. F. DRĂGAN, E. KOHLER, Linear Time Algorithms for Hamiltonian Problems on (Claw, Net)-free Graphs, SIAM J. COMPUT., Vol. 30(5), 2000, pp. 1662-1677.
- BROUSEK, J., Minimal 2-connected Non-Hamiltonian Claw-free Graphs, Discrete Math, vol. 191, 1998, pp. 57-64.
- CHUDNOVSKY, M., P. SEYMOUR, The Structure of Claw-free Graphs, Surveys in Combinatorics, London Math. Soc. Lecture Note Ser. 327, Cambridge: Cambridge Univ. Press, 2005, pp. 153-171.
- CORNEIL, D. G., S. OLARIU, L. STEWART, A Linear Time Algorithm to Compute a Dominating Path in an AT-free Graph, Inform. Proc. Lett. Vol. 54, 1995, pp. 253-257.
- CORNEIL, D. G., S. OLARIU, L. STEWART, Asteroidal Triple-free Graphs, SIAM J. Discrete Math., vol. 10, 1997, pp. 399-430.
- CORNEIL, D. G., S. OLARIU, L. STEWART, Linear Time Algorithms for Dominating Pairs in Asteroidal Triple-free Graphs, SIAM J. Comput. vol. 28, 1999, pp. 1284-1297.
- CROITORU, C., E. OLARU, M. TALMACIU, Confidentially connected Graphs, The annals of “Dunarea de Jos” Univ. of Galati, Proc. of the Intl. Conf. “The Risk in Contemporary Economy”, Suppl. to Tome XVIII (XXII), 2000, pp. 17-20.
- CROITORU, C., M. TĂLMACIU, On Confidentially Connected Graphs, Bul. Stiint. Univ. Baia Mare, Ser B, Matematica – Informatica, vol. XVI(1), 2000, pp. 13-16.
- DAMASCHKE, P., Hamiltonian-hereditary Graphs, unpublished.
- DUFFUS, D., M. S. JACOBSON, R. J. GOULD, Forbidden Subgraphs and the Hamiltonian Theme, in The Theory and Applications of Graphs, (Kalamazoo, Mich, 1980), Wiley, New York, pp. 297-316.
- EDMONDS J., Paths, Trees and Flowers, Canadian Journal of Math, vol. 17, 1965, pp. 449-467.
- EVEN, S., A. PNUELI, A. LEMPEL, Permutation Graphs and Transitive Graphs, J. ACM, vol. 19, 1972, pp. 400-410.
- FLANDRIN, E., J. L. FOUQUET, H. LI, On Hamiltonian Claw-free Graphs, Discrete Math., vol. 111, 1993, pp. 221-229.
- GOLUMBIC, M. C., C. L. MONMA, W. T. TROTTER Jr., Tolerance Graphs, Discrete Applied Mathematics, vol. 9, 1984, pp. 157-170.
- HABIB, M., C. PAUL, A Simple Linear Time Algorithm for Cograph Recognition, Discrete Applied Mathematics, vol. 145, 2005, pp. 183-197.
- HERNANDEZ-MARTINEZ, E. G., E. ARANDA-BRICAIRE, Decentralized Formation Control of Multi-agent Robot Systems based on Formation Graphs, Studies in Informatics and Control, vol. 21(1), pp. 7-16, 2012.
- KLOKS, T., D. KRATSCH, H. MŰLLER, Finding and Counting Small Induced Subgraphs Efficiently, Information Proc. Letters, vol. 74(34), 2000, pp. 115-121
- KOHLER, E., Linear Time Algorithms for Hamiltonian Problems in Claw-Free AT-Free Graphs, manuscript, 1999.
- Las VERGNAS, M. A Note on Matchings in Graphs, Cahiers du Centre d’etudes de Recherche Operationnelle, vol. 17(2-3-4), 1975, pp. 257-260.
- LEKKERKERKER, C. G., J. C. BOLAND, Representation of a Finite Graph by a Set of Intervals on the Real Line, Fund. Math., vol. 51, 1962, pp. 45-64.
- LI, M., Hamiltonian Cycles in 3-Connected Claw-free Graphs, J. Graph Theory, vol. 17, 1993, pp. 303-313.
- MINTY, G. J., On Maximal Independent Sets of Vertices in Claw-free Graphs, Journal of Combinatorial Theory. Series B, vol. 28(3), 1980, pp. 284-304.
- NAKAMURA, D., A. TAMURA, A Revision of Minty’s Algorithm for Finding a Maximum Weighted Stable Set of a Claw-free Graph, Journal of the Operations Research Society of Japan, vol. 44(2), 2001, pp. 194-204.
- POCATILU, P., I. IVAN, A Genetic Algorithm-based System for Automatic Control of Test Data Generation, Studies in Informatics and Control, vol. 22(2), 2013, pp. 219-226.
- SBIHI, N., Algorithme de recherche d’un stable de cardinalite maximum dans un graphe sans etoile, Discrete Mathematics, vol. 29(1), 1980, pp. 53-76.
- SHEPHERD, F. B., Hamiltonicity in Claw-free Graphs, J. Combin. Theory Ser. B, vol. 53, 1991, pp. 173-194.
- SUMNER, D. P., Graphs with 1-factors, Proceedings of the American Mathematical Society (American Mathematical Society), vol. 42(1), 1974, pp. 8-12.
- TALMACIU, M., Decomposition Problems in the Graph Theory with Applications in Combinatorial Optimization – Ph. D. Thesis, “Al. I. Cuza” University Iasi, Romania, 2002.
- TĂLMACIU, M., E. NECHITA, Recognition Algorithm for Diamond-free Graphs, Informatica, vol.18(3), 2007, pp. 457-462.