On the complexity of graph embeddings

Chen, Jianer; Kanchi, Saroja; Kanevsky, Arkady

doi:10.1007/3-540-57155-8_251

Cited by 8 publications

(5 citation statements)

References 7 publications

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“…where ξ(G, T ) denotes the number of components of the cotree G−E(T ) with odd number of edges and the minimum is taken over all spanning trees T of G. A spanning tree reaching this minimum is called a Xuong tree. Every Xuong tree also attains the minimum of formula (1) in Theorem 1 (see for example [1]) but not necessarily vice versa. Examples of graphs with a spanning tree that satisfies (1) but is not a Xuong tree are easy to find.…”

Section: Preliminariesmentioning

confidence: 99%

Matchings, Cycle Bases, and the Maximum Genus of a Graph

Kotrbčík

Škoviera

2012

Electron. J. Combin.

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We study the interplay between the maximum genus of a graph and bases of its cycle space via the corresponding intersection graph. Our main results show that the matching number of the intersection graph is independent of the basis precisely when the graph is upper-embeddable, and completely describe the range of matching numbers when the graph is not upper-embeddable. Particular attention is paid to cycle bases consisting of fundamental cycles with respect to a given spanning tree. For $4$-edge-connected graphs, the intersection graph with respect to any spanning tree (and, in fact, with respect to any basis) has either a perfect matching or a matching missing exactly one vertex. We show that if a graph is not $4$-edge-connected, different spanning trees may lead to intersection graphs with different matching numbers. We also show that there exist $2$-edge connected graphs for which the set of values of matching numbers of their intersection graphs contains arbitrarily large gaps.

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Section: Preliminariesmentioning

confidence: 99%

Matchings, Cycle Bases, and the Maximum Genus of a Graph

Kotrbčík

Škoviera

2012

Electron. J. Combin.

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“…To conclude, we mention the studies of Archdeacon (1990) and Chen et al (1993) on the complexity of graph embedding procedures. It is well-known that embedding a graph G into a surface of minimum genus γ min (G) is NP-hard whereas embedding a graph G into a surface of maximum genus γ max (G) can be done in polynomial time.…”

Section: On the Complexity Of Graph Embeddingmentioning

confidence: 99%

Graph embedding and geometric deep learning relevance to network biology and structural chemistry

Lecca,

Lecca

2023

Front. Artif. Intell.

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Graphs are used as a model of complex relationships among data in biological science since the advent of systems biology in the early 2000. In particular, graph data analysis and graph data mining play an important role in biology interaction networks, where recent techniques of artificial intelligence, usually employed in other type of networks (e.g., social, citations, and trademark networks) aim to implement various data mining tasks including classification, clustering, recommendation, anomaly detection, and link prediction. The commitment and efforts of artificial intelligence research in network biology are motivated by the fact that machine learning techniques are often prohibitively computational demanding, low parallelizable, and ultimately inapplicable, since biological network of realistic size is a large system, which is characterised by a high density of interactions and often with a non-linear dynamics and a non-Euclidean latent geometry. Currently, graph embedding emerges as the new learning paradigm that shifts the tasks of building complex models for classification, clustering, and link prediction to learning an informative representation of the graph data in a vector space so that many graph mining and learning tasks can be more easily performed by employing efficient non-iterative traditional models (e.g., a linear support vector machine for the classification task). The great potential of graph embedding is the main reason of the flourishing of studies in this area and, in particular, the artificial intelligence learning techniques. In this mini review, we give a comprehensive summary of the main graph embedding algorithms in light of the recent burgeoning interest in geometric deep learning.

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“…The following useful lemma, found for example in [4], is an extension of Lemma 2 to embeddings with more than one face. It can either be proved directly by using Ringeisen's edge-adding technique or can be derived from Xuongs's theorem.…”

Section: Introductionmentioning

confidence: 99%

“…For graphs of bounded maximum degree [2] has recently proposed a polynomial-time algorithm constructing an embedding with genus at most O(γ(G) c 1 log c 2 n) where c 1 and c 2 are constants. On the other hand, for every ǫ > 0 and every function f (n) = O(n 1−ǫ ) there is no polynomial-time algorithm that constructs an embedding of any graph G with n vertices into the surface of genus at most γ(G) + f (n) unless P = NP (see [4,5]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Simple greedy 2-approximation algorithm for the maximum genus of a graph

Kotrbcik,

Skoviera

2015

Preprint

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The maximum genus γ M (G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal results in a connected spanning subgraph of G.In this paper we prove that removing pairs of adjacent edges from G arbitrarily while retaining connectedness leads to at least γ M (G)/2 pairs of edges removed. This allows us to describe a greedy algorithm for the maximum genus of a graph; our algorithm returns an integer k such that γ M (G)/2 ≤ k ≤ γ M (G), providing a simple method to efficiently approximate maximum genus. As a consequence of our approach we obtain a 2-approximate counterpart of Xuong's combinatorial characterisation of maximum genus.

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On the complexity of graph embeddings

Cited by 8 publications

References 7 publications

Matchings, Cycle Bases, and the Maximum Genus of a Graph

Matchings, Cycle Bases, and the Maximum Genus of a Graph

Graph embedding and geometric deep learning relevance to network biology and structural chemistry

Simple greedy 2-approximation algorithm for the maximum genus of a graph

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