Abstract
This paper focuses on two themes within the broad context of recursively definable graph classes. First, we generalize the series-parallel operations and establish exactly how far they can be extended subject to some consistency conditions. We show explicitly how Halin graphs are included in the extension. Second, for recursively constructed graphs in general, we construct a predicate calculus within which graph problems can be stated and for those so stated, a linear time algorithm exists and can be automatically generated. We discuss some issues related to practical automatic generation.
Similar content being viewed by others
References
S. Arnborg, Efficient algorithms for combinatorial problems on graphs with bounded decomposability, BIT 25(1985)2–23.
S. Arnborg, D.G. Corneil and A. Proskurowski, Complexity of finding embeddings in ak-tree, SIAM J. Alg. Discr. Meth. 8(1987)277–284.
S. Arnborg, J. Lagergren and D. Seese, Problems easy for tree-decomposable graphs, J. Algorithms, to appear.
S. Arnborg and A. Proskurowski, Characterization and recognition of partialk-trees, TRITA-NA-8402, Royal Institute of Technology, Sweden (1984).
S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems on graphs embedded ink-trees, TRITA-NA-8404, Royal Institute of Technology, Sweden (1984).
M. Bauderon and B. Courcelle, Graph expressions and graph rewritings, Math. Syst. Theory 20(1987)83–127.
M.W. Bern, E.L. Lawler and A.L. Wong, Linear time computation of optimal subgraphs of decomposable graphs, J. Algorithms 8(1987)216–235.
H.L. Bodlaender, Dynamic programming on graphs with bounded tree-width, Technical Report, Laboratory for Computer Science, MIT, (1987); extended abstract inProc. ICALP (1988).
R.B. Borie, R.G. Parker and C.A. Tovey, Generalizedk-jackknife operations and partialk-trees, Technical Report No. J-88-13, School of ISyE, Georgia Tech. (1988).
R.B. Borie, R.G. Parker and C.A. Tovey, Unambiguous factorization of recursive graph classes, Technical Report No. J-88-15, School of ISyE, Georgia Tech. (1988), SIAM J. Discr. Math., submitted.
R.B. Borie, R.G. Parker and C.A. Tovey, Automatic generation of linear algorithms from predicate calculus descriptions of problems on recursively constructed graph families, Technical Report No. J-88-16, School of ISyE, Georgia Tech. (1988), Algorithmica, to appear.
B. Courcelle, Recognizability and second-order definability for sets of finite graphs, Université de Bordeaux, I-8634 (1987).
R.J. Duffin, Topology of series-parallel networks, J. Math. Anal. Appl. 10(1965)303–318.
E. Hare, S. Hedetniemi, R. Laskar, K. Peters and T.V. Wimer, Linear-time computability of combinatorial problems on generalized series-parallel graphs, Discr. Algorithms and Complexity (1987)437–457.
S. Mahajan and J.G. Peters, Algorithms for regular properties in recursive graphs,25th Annual Allerton Conf. on Communications, Control, and Computing (1987), pp. 14–23.
R.L. Rardin, R.G. Parker and D.K. Wagner, Definitions, properties, and algorithms for detecting series-parallel graphs, Technical Report, Department of Industrial Engineering, Purdue University (1982).
M.B. Richey, Combinatorial optimization on series-parallel graphs: Algorithms and complexity, Ph.D. Thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology (1985).
M.B. Richey, R.G. Parker and R.L. Rardin, On finding spanning Eulerian subgraphs, Naval Res. Logist. Quart. 32(1985)443–455.
M.B. Richey and R.G. Parker, Minimum-maximal matching in series-parallel graphs, Eur. J. Oper. Res. 33(1987)98–105.
M.B. Richey and R.G. Parker, On multiple Steiner subgraph problems, Networks 16(1986)423–438.
M.B. Richey, R.G. Parker and R.L. Rardin, An efficiently solvable case of the minimum weight equivalent subgraph problem, Networks 15(1985)217–228.
K. Takamizawa, T. Nishizeki and N. Saito, Linear-time computability of combinatorial problems on series-parallel graphs, J. ACM 29(1982)623–641.
J.A. Wald and C.J. Colbourn, Steiner trees, partial 2-trees, and minimum IFI networks, Networks 13(1983)159–167.
T.V. Wimer, Linear algorithms onk-terminal graphs, Ph.D. Thesis, Report No. URI-030, Clemson University (1987).
T.V. Wimer and S.T. Hedetniemi,K-terminal recursive families of graphs,Proc. 250th Anniversary Conf. on Graph Theory (1986).
T.V. Wimer, S.T. Hedetniemi and R. Laskar, A methodology for constructing linear graph algorithms, Congressus Numerantium 50(1985)43–60.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Borie, R., Parker, R.G. & Tovey, C.A. Algorithms for recognition of regular properties and decomposition of recursive graph families. Ann Oper Res 33, 125–149 (1991). https://doi.org/10.1007/BF02115752
Issue Date:
DOI: https://doi.org/10.1007/BF02115752