# On dually compact closed classes of graphs and BFS-constructible graphs

Discussiones Mathematicae Graph Theory (2003)

- Volume: 23, Issue: 2, page 365-381
- ISSN: 2083-5892

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topNorbert Polat. "On dually compact closed classes of graphs and BFS-constructible graphs." Discussiones Mathematicae Graph Theory 23.2 (2003): 365-381. <http://eudml.org/doc/270651>.

@article{NorbertPolat2003,

abstract = {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.},

author = {Norbert Polat},

journal = {Discussiones Mathematicae Graph Theory},

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; dually compact closed; chordal graphs; bridged graphs; infinite graphs; BFS-constructible graph; modular graphs; Helly graphs; pseudo-median graphs},

language = {eng},

number = {2},

pages = {365-381},

title = {On dually compact closed classes of graphs and BFS-constructible graphs},

url = {http://eudml.org/doc/270651},

volume = {23},

year = {2003},

}

TY - JOUR

AU - Norbert Polat

TI - On dually compact closed classes of graphs and BFS-constructible graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2003

VL - 23

IS - 2

SP - 365

EP - 381

AB - 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.

LA - eng

KW - 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; dually compact closed; chordal graphs; bridged graphs; infinite graphs; BFS-constructible graph; modular graphs; Helly graphs; pseudo-median graphs

UR - http://eudml.org/doc/270651

ER -

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