Minimal extensions of graphs to absolute retracts

Pesch, Erwin

doi:10.1002/jgt.3190110416

Cited by 18 publications

(9 citation statements)

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“…The second property that we need is the existence of an injective hull for any finite metric space over N and in particular for every finite graph. This result was obtained by Isbell [6], by Pouzet [14,15] (see also [8,) and also by Pesch [10]. Lemma 5.4 ([6], [14,15] and [10]).…”

Section: N Polatmentioning

confidence: 60%

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

Polat

2003

Discuss. Math. Graph Theory

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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 wellordering ≤ (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 intervalfinite 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 366 N. Polat (resp. Helly) subgraph of G. We also give two sufficient conditions so that a bridged graph has a similar property.

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Section: N Polatmentioning

confidence: 60%

“…This result was obtained by Isbell [6], by Pouzet [14,15] (see also [8,) and also by Pesch [10]. Lemma 5.4 ([6], [14,15] and [10]). For every finite metric space (E, d) over N there exists, up to isomorphism, a unique finite minimal Helly graph G (injective hull) such that (E, d) is an isometric subspace of (V (G), d G ).…”

Section: N Polatmentioning

confidence: 60%

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

Polat

2003

Discuss. Math. Graph Theory

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“…Also, due to results of Pesch [14] and of Jawhari, Misane, and Pouzet [12, Theorem IV-1.2.2], the class of absolute retracts of reflexive graphs, or Helly graphs, is dually compact closed. Hahn, Sauer, and Woodrow [9] suggested to study the dually compact closed classes of graphs, and in particular to determine if the class of bridged graphs is dually compact closed.…”

Section: Claim 42 G Is An Isometric Subgraph Of Gmentioning

confidence: 96%

On constructible graphs, locally Helly graphs, and convexity

Polat

2003

Journal of Graph Theory

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A (finite or infinite) graph G is constructible if there exists a well‐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. Particular constructible graphs are Helly graphs and connected bridged graphs. In this paper we study a new class of constructible graphs, the class of locally Helly graphs. A graph G is locally Helly if, for every pair (x,y) of vertices of G whose distance is d ≥ 2, there exists a vertex whose distance to x is d − 1 and which is adjacent to y and to all neighbors of y whose distance to x is at most d. Helly graphs are locally Helly, and the converse holds for finite graphs. Among different properties we prove that a locally Helly graph is strongly dismantable, hence cop‐win, if and only if it contains no isometric rays. We show that a locally Helly graph G is finitely Helly, that is, every finite family of pairwise non‐disjoint balls of G has a non‐empty intersection. We give a sufficient condition by forbidden subgraphs so that the three concepts of Helly graphs, of locally Helly graphs and of finitely Helly graphs are equivalent. Finally, generalizing different results, in particular those of Bandelt and Chepoi 1 about Helly graphs and bridged graphs, we prove that the Helly number h(G) of the geodesic convexity in a constructible graph G is equal to its clique number ω(G), provided that ω(G) is finite. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 280–298, 2003

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“…The concept of strong isometric dimension has two motivations. It is implicit in the concept of an injective hull of a graph [6,7,10,13]. The injective hull of a graph G is the smallest supergraph H of G where H is an absolute retract, i.e., H is a retract of a graph whenever it is an induced subgraph of that graph.…”

Section: Introductionmentioning

confidence: 99%