PDF

Discussiones Mathematicae Graph Theory  18(1) (1998)   23-48
DOI: https://doi.org/10.7151/dmgt.1061

THE LEAFAGE OF A CHORDAL GRAPH

In-Jen Lin National Ocean University, Taipei, Taiwan

Terry A. McKee Wright State University, Dayton, OH 45435-0001, U.S.A

Douglas B. West University of Illinois, Urbana, IL 61801-2975, U.S.A.

Abstract

The leafage l(G) of a chordal graph G is the minimum number of leaves of a tree in which G has an intersection representation by subtrees. We obtain upper and lower bounds on l(G) and compute it on special classes. The maximum of l(G) on n-vertex graphs is n - lg n -1 lg lg n + O(1). The proper leafage l^*(G) is the minimum number of leaves when no subtree may contain another; we obtain upper and lower bounds on l^*(G). Leafage equals proper leafage on claw-free chordal graphs. We use asteroidal sets and structural properties of chordal graphs.

Keywords: chordal graph, subtree intersection representation, leafage.

1991 Mathematics Subject Classification: 05C75, 05C05, 05C35.

References

[1]	H. Broersma, T. Kloks, D. Kratsch and H. Müller, Independent sets in asteroidal triple-free graphs, in: Proceedings of ICALP'97, P. Degano, R. Gorrieri, A. Marchetti-Spaccamela, (eds.), (Springer-Verlag, 1997), Lect. Notes Comp. Sci. 1256, 760-770.
[2]	H. Broersma, T. Kloks, D. Kratsch and H. Müller, A generalization of AT-free graphs and a generic algorithm for solving triangulation problems, Memorandum No. 1385, University of Twente, Enschede, The Netherlands, 1997.
[3]	P.A. Buneman, A characterization of rigid circuit graphs, Discrete Math. 9 (1974) 205-212, doi: 10.1016/0012-365X(74)90002-8.
[4]	R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math. 51 (1950) 161-166, doi: 10.2307/1969503.
[5]	R.P. Dilworth, Some combinatorial problems on partially ordered sets, in: Combinatorial Analysis (Bellman and Hall, eds.) Proc. Symp. Appl. Math. (Amer. Math. Soc 1960), 85-90.
[6]	G.A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71-76, doi: 10.1007/BF02992776.
[7]	D.R. Fulkerson and O.A. Gross, Incidence matrices and interval graphs, Pac. J. Math. 15 (1965) 835-855.
[8]	F. Gavril, The intersection graphs of subtrees in trees are exactly the chordal graphs, J. Combin. Theory (B) 16 (1974) 47-56, doi: 10.1016/0095-8956(74)90094-X.
[9]	F. Gavril, Generating the maximum spanning trees of a weighted graph, J. Algorithms 8 (1987) 592-597, doi: 10.1016/0196-6774(87)90053-8.
[10]	P.C. Gilmore and A.J. Hoffman, A characterization of comparability graphs and of interval graphs, Canad. J. Math. 16 (1964) 539-548, doi: 10.4153/CJM-1964-055-5.
[11]	M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, 1980).
[12]	T. Kloks, D. Kratsch and H. Müller, Asteroidal sets in graphs, in: Proceedings of WG'97, R. Möhring, (ed.), (Springer-Verlag, 1997) Lect. Notes Comp. Sci. 1335, 229-241.
[13]	T. Kloks, D. Kratsch and H. Müller, On the structure of graphs with bounded asteroidal number, Forschungsergebnisse Math/Inf/97/22, FSU Jena, Germany, 1997.
[14]	B. Leclerc, Arbres et dimension des ordres, Discrete Math. 14 (1976) 69-76, doi: 10.1016/0012-365X(76)90007-8.
[15]	C.B. Lekkerkerker and J.Ch. Boland, Representation of a finite graph by a set of intervals on the real line, Fund. Math. 51 (1962) 45-64.
[16]	I.-J. Lin, M.K. Sen and D.B. West, Leafage of directed graphs, to appear.
[17]	T.A. McKee, Subtree catch graphs, Congr. Numer. 90 (1992) 231-238.
[18]	E. Prisner, Representing triangulated graphs in stars, Abh. Math. Sem. Univ. Hamburg 62 (1992) 29-41, doi: 10.1007/BF02941616.
[19]	F.S. Roberts, Indifference graphs, in: Proof Techniques in Graph Theory (F. Harary, ed.), Academic Press (1969) 139-146.
[20]	D.J. Rose, Triangulated graphs and the elimination process, J. Math. Ann. Appl. 32 (1970) 597-609, doi: 10.1016/0022-247X(70)90282-9.
[21]	D.J. Rose, R.E. Tarjan and G.S. Leuker, Algorithmic aspects of vertex elimination on graphs, SIAM J. Comp. 5 (1976) 266-283, doi: 10.1137/0205021.
[22]	Y. Shibata, On the tree representation of chordal graphs, J. Graph Theory 12 (1988) 421-428, doi: 10.1002/jgt.3190120313.
[23]	J.R. Walter, Representations of chordal graphs as subtrees of a tree, J. Graph Theory 2 (1978) 265-267, doi: 10.1002/jgt.3190020311.

Received 2 January 1997
Revised 19 April 1998