Compatible topologies on graphs: An application to graph isomorphism problem complexity

Bretto, Alain; Faisant, Alain; Vallée, Thierry

doi:10.1016/j.tcs.2006.07.010

Cited by 5 publications

(8 citation statements)

References 14 publications

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“…Then f need not be a K-continuous map. More precisely, assume SC 3,4 26 := (c i ) i∈[0,3] Z as a subspace of (Z 3 , T 3 ) (see Figure 1(b)) and the self-map f : SC 3,4 26 → SC 3,4 26 given by f (c i ) = c i+1(mod 4) . Then we can clearly observe that f cannot be a K-continuous (3) Let us consider the map f : (Z, T) → (Z, T) given by f (t) = t + 1 which is a parallel translation with an odd vector.…”

Section: Remark 33 (1) Letmentioning

confidence: 99%

“…Let us now prove that the converse does not hold in terms of the following example. Consider the space SC n, 4 3 n −1 := (c i ) i∈[0,3] Z in which two points are pure open and the others are pure closed, and the self-map f : SC n, 4 3 n −1 → SC n, 4 3 n −1 given by f (c i ) = c i+1(mod 4) . Then we can observe that f is (3 n − 1)-continuous from the viewpoint of CTC.…”

Section: Definition 34 ([13]mentioning

confidence: 99%

“…In this regard, both a Khalimsky continuous map and a Khalimsky homeomorphism have been often used in digital geometry [5, 6, 13, 18, 20-22, 24, 28, 31]. But it is well known that the graph k-connectivity of Z n and every topological connectedness as well as Khalimsky connectedness are partially compatible with each other [4,5,25]. Besides, a Khalimsky continuous map has some limitations of performing a discrete geometric transformation such as a rotation by 90 • and a parallel translation with an odd vector (see Remark 3.3).…”

Section: Introductionmentioning

confidence: 99%

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Compression of Khalimsky topological spaces

Kang

Han

2012

Filomat

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Aiming at the study of the compression of Khalimsky topological spaces which is an interesting field in digital geometry and computer science, the present paper develops a new homotopy thinning suitable for the work. Since Khalimsky continuity of maps between Khalimsky topological spaces has some limitations of performing a discrete geometric transformation, the paper uses another continuity (see Definition 3.4) that can support the discrete geometric transformation and a homotopic thinning suitable for studying Khalimsky topological spaces. By using this homotopy, we can develop a new homotopic thinning for compressing the spaces and can write an algorithm for compressing 2D Khalimsky topological spaces.

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Section: Remark 33 (1) Letmentioning

confidence: 99%

Section: Definition 34 ([13]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Compression of Khalimsky topological spaces

Kang

Han

2012

Filomat

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“…Because these problems are of great importance in scientific research and practical application, there are numerous scientists who consider the graph isomorphism issue to be their primary research area [9]. They try to solve the graph isomorphism problem by using various methods in different fields [3,7,12,13].…”

Section: Introductionmentioning

confidence: 99%

The SVE Method for Regular Graph Isomorphism Identification

Shang

Feng

Chen

et al. 2015

Circuits Syst Signal Process

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A regular graph is a type of graph that has particular features. Such graphs have great symmetry, which increases the difficulty of isomorphism identification. In this paper, we propose a new method to address regular graph isomorphism identification. We make some adjustments to the original circuit simulation (CS) method in the graph isomorphism identification process, using single-vertex excitation (SVE) in place of the original excitation, which could remove the confusion of corresponding vertices caused by the graph's symmetry. This new SVE method could widen the range of the application of the original CS method and create better performance in time consumption compared with other traditional methods. The performance of the SVE method is provided in this paper illuminating its practicality and superiority.

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“…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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Compatible topologies on graphs: An application to graph isomorphism problem complexity

Cited by 5 publications

References 14 publications

Compression of Khalimsky topological spaces

Compression of Khalimsky topological spaces

The SVE Method for Regular Graph Isomorphism Identification

Digital Topologies on Graphs

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