Abstract
The use of (generalized) tree structure in graphs is one of the main topics in the field of efficient graph algorithms. The well-known partial κ-tree (resp. treewidth) approach belongs to this kind of research and bases on a tree structure of constant-size bounded maximal cliques. Without size bound on the cliques this tree structure of maximal cliques characterizes chordal graphs which are known to be important also in connection with relational database schemes where hypergraphs with tree structure (acyclic hypergraphs) and their elimination orderings (perfect elimination orderings for chordal graphs, Graham-reduction for acyclic hypergraphs) are studied.
We consider here graphs with a tree structure which is dual (in the sense of hypergraphs) to that one of chordal graphs (therefore we call these graphs dually chordal). The corresponding vertex elimination orderings of these graphs are the maximum neighbourhood orderings. These orderings were studied recently in several papers and some of the algorithmic consequences of such orderings were given.
The aim of this paper is a systematic treatment of the algorithmic use of maximum neighbourhood orderings. These orderings are useful especially for dominating-like problems (including Steiner tree) and distance problems. Many problems efficiently solvable for strongly chordal graphs remain efficiently solvable for dually chordal graphs too.
Our results on dually chordal graphs not only generalize, but also improve and extend the corresponding results on strongly chordal graphs, since a maximum neighbourhood ordering (if it exists) can be constructed in linear time and we consequently use the underlying structure properties of dually chordal graphs closely connected to hypergraphs.
Second and third author supported by the VW-Stiftung Project No. I/69041
Preview
Unable to display preview. Download preview PDF.
References
H. Behrendt and A. Brandstädt, Domination and the use of maximum neighbourhoods, Technical Report SM-DU-204, University of Duisburg 1992
C. Berge, Hypergraphs, North Holland, 1989
A. Brandstädt, F.F. Dragan, V.D. Chepoi, and V.I. Voloshin, Dually chordal graphs, Technical Report SM-DU-225, University of Duisburg 1993, Graph-Theoretic Concepts in Computer Science, 19th International Workshop WG'93, Utrecht, The Netherlands, LNCS 790, Springer, Jan van Leeuwen (Ed.), 237–251
A. Brandstädt, V.D. Chepoi, and F.F. Dragan, The algorithmic use of hypertree structure and maximum neighbourhood orderings, Technical Report SM-DU-244, University of Duisburg 1994, submitted to Theor. Comp. Science
P. Buneman, A characterization of rigid circuit graphs, Discr. Math. 9 (1974), 205–212
R. Chandrasekaran and A. Doughety, Location on tree networks: p-center and q-dispersion problems, Math. Oper. Res. 1981, 6, No. 1, 50–57
G.J. Chang, Labeling algorithms for domination problems in sun-free chordal graphs, Discrete Applied Mathematics 22 (1988/89), 21–34
G.J. Chang and G.L. Nemhauser, The κ-domination and κ-stability problems on sun-free chordal graphs, SIAM J. Algebraic and Discrete Methods 5 (1984), 332–345
G.J. Chang and G.L. Nemhauser, Covering, Packing and Generalized Perfection, SIAM J. Algebraic and Discrete Methods 6 (1985), 109–132
F.F. Dragan, Dominating and packing in triangulated graphs (in Russian), Meth. of Disci. Analysis (Novosibirsk) 51 (1991), 17–36
F.F. Dragan, HT-graphs: centers, connected r-domination and Steiner trees, Computer Science Journal of Moldova, 1993, Vol. 1, No. 2, 64–83
F.F. Dragan, Domination in Helly graphs without quadrangles (in Russian) Cybernetics and System Analysis (Kiev) 6 (1993)
F. F. Dragan, and A. Brandstädt, r-Dominating cliques in Helly graphs and chordal graphs, Technical Report SM-DU-228, University of Duisburg 1993, Proc. of the 11th STAGS, Caen, France, Springer, LNCS 775, 735–746, 1994
F. F. Dragan, C. F. Prisacaru, and V. D. Chepoi, Location problems in graphs and the Helly property (in Russian), Discrete Mathematics, Moscow, 4 (1992), 67–73 (the full version appeared as preprint: F.F. Dragan, C.F. Prisacaru, and V.D. Chepoi, r-Domination and p-center problems on graphs: special solution methods and graphs for which this method is usable (in Russian), Kishinev State University, preprint MoldNIINTI, N. 948-M88, 1987)
M. Farber, Domination, Independent Domination and Duality in Strongly Chordal Graphs, Discr. Appl. Math.7 (1984), 115–130
G.N. Frederickson, Parametric search and locating supply centers in trees, Proc. Workshop on Algorithms and Data Structures (WADS'91), Springer, LNCS 519, 1991, 299–319
Y. Gurevich, L. Stockmeyer and U. Vishkin, Solving NP-hard problems on graphs that are almost trees and an application to facility location problems, J. ACM 31 (1984), 459–473
S.C. Hedetniemi and R. Laskar, (eds.), Topics on Domination, Annals of Discr. Math. 48, North-Holland, 1991
O. Kariv and S.L. Hakimi, An algorithmic approach to network location problems, I: the p-centers, SIAM J. Appl. Math. 1979, 37, No. 3, 513–538
A.W.J. Kolen, Duality in tree location theory, Cah. Cent. Etud. Rech. Oper., 25 (1983), 201–215
A. Lubiw, Doubly lexical orderings of matrices, SIAM J. Comput. 16 (1987), 854–879
N. Megiddo, A. Tamir, E. Zemel and R. Chandrasekaran, An O(nlog2n) algorithm for the κ-th longest path in a tree with applications to location problems, SIAM J. Comput. 10 (1981), 328–337
M. Moscarini, Doubly chordal graphs, Steiner trees and connected domination, Networks 23 (1993), 59–69
R. Paige and R.E. Tarjan, Three partition refinement algorithms, SIAM J. Comput. 16 (1987), 973–989
J. Plesnik, A heuristic for the p-center problem in graphs, Discr. Math. 17 (1987), 263–268
P.J. Slater, R-domination in graphs, J. ACM 23 (1976), 446–450
J.P. Spinrad, Doubly lexical ordering of dense 0–1-matrices, manuscript 1988, to appear in SIAM J. Comput.
J.L. Szwarcfiter and C.F. Bornstein, Clique graphs of chordal and path graphs, manuscript 1992, to appear in SIAM J. Discr. Math.
B. Tansel, R. Francis and T. Lowe, Location on networks: a survey I, II, Management Sci. 29 (1983), 482–511
R.E. Tarjan and M. Yannakakis, Simple linear time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J. Comput. 13, 3 (1984), 566–579
K. White, M. Farber and W. Pulleyblank, Steiner Trees, Connected Domination and Strongly Chordal Graphs, Networks 15 (1985), 109–124
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Brandstädt, A., Chepoi, V.D., Dragan, F.F. (1995). The algorithmic use of hypertree structure and maximum neighbourhood orderings. In: Mayr, E.W., Schmidt, G., Tinhofer, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 1994. Lecture Notes in Computer Science, vol 903. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59071-4_38
Download citation
DOI: https://doi.org/10.1007/3-540-59071-4_38
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59071-2
Online ISBN: 978-3-540-49183-5
eBook Packages: Springer Book Archive