ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 23(2) (2003) 365-381
DOI: 10.7151/dmgt.1207


Norbert Polat

I.A.E., Université Jean Moulin - Lyon 3
6, cours Albert Thomas - B.P. 8242
69355 Lyon Cedex 08, France

Dedicated to Wilfried Imrich on the occasion of his 60th birthday.


A class C of graphs is said to be dually compact closed if, for every infinite G ∈ C, each finite subgraph of G is contained in a finite induced subgraph of G which belongs to C. The class of trees and more generally the one of chordal graphs are dually compact closed. One of the main part of this paper is to settle a question of Hahn, Sands, Sauer and Woodrow by showing that the class of bridged graphs is dually compact closed. To prove this result we use the concept of constructible graph. A (finite or infinite) graph G is constructible if there exists a well-ordering ≤ (called constructing ordering) of its vertices such that, for every vertex x which is not the smallest element, there is a vertex y < x which is adjacent to x and to every neighbor z of x with z < x. Finite graphs are constructible if and only if they are dismantlable. The case is different, however, with infinite graphs. A graph G for which every breadth-first search of G produces a particular constructing ordering of its vertices is called a BFS-constructible graph. We show that the class of BFS-constructible graphs is a variety (i.e., it is closed under weak retracts and strong products), that it is a subclass of the class of weakly modular graphs, and that it contains the class of bridged graphs and that of Helly graphs (bridged graphs being very special instances of BFS-constructible graphs). Finally we show that the class of interval-finite pseudo-median graphs (and thus the one of median graphs) and the class of Helly graphs are dually compact closed, and that moreover every finite subgraph of an interval-finite pseudo-median graph (resp. a Helly graph) G is contained in a finite isometric pseudo-median (resp. Helly) subgraph of G. We also give two sufficient conditions so that a bridged graph has a similar property.

Keywords: infinite graph; dismantlable graph; constructible graph; BFS-cons-tructible graph; variety; weak-retract; strong product; bridged graph; Helly graph; weakly-modular graph; dually compact closed class.

2000 Mathematics Subject Classification: 05C75, 05C99.


[1]R.P. Anstee and M. Farber, On bridged graphs and cop-win graphs, J. Combin. Theory (B) 44 (1988) 22-28, doi: 10.1016/0095-8956(88)90093-7.
[2]M. Chastand, F. Laviolette and N. Polat, On constructible graphs, infinite bridged graphs and weakly cop-win graphs, Discrete Math. 224 (2000) 61-78, doi: 10.1016/S0012-365X(00)00127-8.
[3]G. Hahn, F. Laviolette, N. Sauer and R.E. Woodrow, On cop-win graphs, preprint, 1989.
[4]G. Hahn, N. Sands, N. Sauer and R.E. Woodrow, Problem Session, in: Colloque Franco-Canadien de Combinatoire (Université de Montréal, 1981).
[5]G. Hahn, N. Sauer and R.E. Woodrow, personal communication.
[6]J. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65-76, doi: 10.1007/BF02566944.
[7]W. Imrich and S. Klavžar, Product graphs (Wiley, 2000).
[8]E. Jawhari, D. Misane and M. Pouzet, Retracts: Graphs and ordered sets from the metric point of view, Contemp. Math. 57 (1986) 175-226, doi: 10.1090/conm/057/856237.
[9]R. Nowakowski and I. Rival, The smallest graph variety containing all paths, Discrete Math. 43 (1983) 223-234, doi: 10.1016/0012-365X(83)90159-0.
[10]E. Pesch, Minimal extensions of graphs to absolute retracts, J. Graph Theory 11 (1987) 585-598, doi: 10.1002/jgt.3190110416.
[11]N. Polat, Invariant subgraph properties in pseudo-modular graphs, Discrete Math. 207 (2000) 199-217, doi: 10.1016/S0012-365X(99)00045-X.
[12]N. Polat, On infinite bridged graphs and strongly dismantlable graphs, Discrete Math. 211 (2000) 153-166, doi: 10.1016/S0012-365X(99)00142-9.
[13]N. Polat, On isometric subgraphs of infinite bridged graphs and geodesic convexity, Discrete Math. 244 (2002) 399-416, doi: 10.1016/S0012-365X(01)00097-8.
[14]M. Pouzet, Une approche métrique de la rétraction dans les ensembles ordonnés et les graphes, Compte-rendu des Journées infinitistes de Lyon (octobre 1984), Pub. Dépt. Math. (Lyon, 1985).
[15]M. Pouzet, Retracts: recent and old results on graphs, ordered sets and metric spaces, Circulating manuscript, 29 pages, Nov. 1983.
[16]A. Quilliot, Homomorphismes, points fixes, rétractions et jeux de poursuite dans les graphes, les ensembles ordonnés et les espaces métriques, Thrèse de doctorat d'Etat (Univ. Paris VI, 1983).
[17]V. Soltan and V. Chepoi, Conditions for invariance of set diameters under d-convexification, Cybernetics 19 (1983) 750-756, doi: 10.1007/BF01068561.
[18]C. Tardif, On compact median graphs, J. Graph Theory 23 (1996) 325-336, doi: 10.1002/(SICI)1097-0118(199612)23:4<325::AID-JGT1>3.0.CO;2-T.

Received 26 September 2001
Revised 6 May 2002