Graphs and topologies on discrete sets

Préa, Pascal

doi:10.1016/0012-365x(92)90269-l

Cited by 7 publications

(4 citation statements)

References 6 publications

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“…Préa introduced the notion of compatibility between a graph G and a topology on its vertex set V (G) in [7]. There, a topology τ on V (G) was said to be compatible (resp., weakly compatible) with a graph G if every induced subgraph H of G (resp., every finite induced subgraph H of G) is connected if and only if the relative topology induced by τ on V (H) is connected.…”

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Compact compatible topologies for graphs with small cycles

Neumann-Lara¹,

Wilson²

2005

International Journal of Mathematics and Mathematical Sciences

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A topology τ on the vertices of a comparability graph G is said to be compatible with G if each subgraph H of G is graph-connected if and only if it is a connected subspace of (G,τ). In two previous papers we considered the problem of finding compatible topologies for a given comparability graph and we proved that the nonexistence of infinite paths was a necessary and sufficient condition for the existence of a compact compatible topology on a tree (that is to say, a connected graph without cycles) and we asked whether this condition characterized the existence of a compact compatible topology on a comparability graph in which all cycles are of length at most n. Here we prove an extension of the above-mentioned theorem to graphs whose cycles are all of length at most five and we show that this is the best possible result by exhibiting a comparability graph in which all cycles are of length 6, with no infinite paths, but which has no compact compatible topology.

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confidence: 99%

“…Such a topology will be said to be consistent with the poset or ordered comparability graph. Clearly a consistent topology must be weakly compatible in the sense of [5,7], but not vice versa. In what follows, in order to avoid unnecessary terminology, we assume that the conditions of weak compatibility and compatibility include the condition of consistency.…”

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confidence: 99%

Compact compatible topologies for graphs with small cycles

Neumann-Lara¹,

Wilson²

2005

International Journal of Mathematics and Mathematical Sciences

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“…Recall, [15], that a topology T on the set of vertices V of a graph G = (V ; E) is compatible if the connected subspaces of (V, T ) are the same as the connected induced subgraphs of G = (V ; E).…”

Section: Compatible Topologies On Graphmentioning

confidence: 99%

“…From this definition some questions arise about the continuity of f or f −1 , the structural properties of f , etc. Consequently compatible topologies have been intensively studied and there are a lot of contributions by many authors [5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Digital Topologies on Graphs

Bretto¹

Applied Graph Theory in Computer Vision and Pattern Recognition

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Summary. In this chapter we focus on the relationship between graph theory and topology. Topologies on vertices of graphs became much more essential in topology, with the development of computer science, especially with the development of computer graphic and image analysis. Digital topology is the study of the topological properties of digital images. In most of the literature a digital image has been endowed with a graph model; the vertices being the points of the image, and the edges giving the connectivity between the points. This has led to the investigation of topology on graph [7][8][9][10][11][12]. We study compatible topologies on graphs, (here compatibility is to be understood as connectivity). We describe some properties of these particular topological spaces. We discuss the relation between T0-spaces (which play an important role) and other compatible topologies on graph. We develop some applications to digital geometry. Other results related to compatible topologies on graphs is developed.

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Comparability Graphs and Digital Topology

Bretto¹

2001

Computer Vision and Image Understanding

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Graphs and topologies on discrete sets

Cited by 7 publications

References 6 publications

Compact compatible topologies for graphs with small cycles

Compact compatible topologies for graphs with small cycles

Digital Topologies on Graphs

Comparability Graphs and Digital Topology

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