Four-terminal reducibility and projective-planar wye-delta-wye-reducible graphs

Archdeacon, Dan; Colbourn, Charles J.; Gitler, Isidoro; Provan, J. Scott

doi:10.1002/(sici)1097-0118(200002)33:2<83::aid-jgt3>3.0.co;2-p

Cited by 19 publications

(33 citation statements)

References 17 publications

(27 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Like Gitler [27], Feo and Provan [24], and Archdeacon et al [7], we conjecture that any n-vertex plane graph can be reduced to a vertex using O(n 3/2 ) facial electrical transformations. More ambitiously, we conjecture an upper bound of O(nt) for any n-vertex plane graph with treewidth t.…”

Section: Cylindrical Gridssupporting

confidence: 55%

“…The cylindrical grid graphs C(4, 7) and C(3,8) and (in light gray) their medial graphs T (8, 7) and T(7,8).• C (p, q ) is obtained by connecting a new vertex to the vertices of one of the q-gonal faces of C(p, q), or equivalently, by contracting one of the q-gonal faces of C(p + 1, q) to a single vertex. If q is even, then the medial graph of C (p, q) is the flat torus knot T (2p + 1, q).…”

mentioning

confidence: 99%

See 1 more Smart Citation

Untangling Planar Curves

Chang

Erickson

2017

Discrete Comput Geom

View full text Add to dashboard Cite

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Θ(n 3/2 ) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n 2 ), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. Our lower bound also implies that Ω(n 3/2 ) facial electrical transformations are required to reduce any plane graph with treewidth Ω( n) to a single vertex, matching known upper bounds for rectangular and cylindrical grid graphs. More generally, we prove that transforming one immersion of k circles with at most n self-crossings into another requires Θ(n 3/2 + nk + k 2 ) homotopy moves in the worst case. Finally, we prove that transforming one noncontractible closed curve to another on any orientable surface requires Ω(n 2 ) homotopy moves in the worst case; this lower bound is tight if the curve is homotopic to a simple closed curve.

show abstract

Section: Cylindrical Gridssupporting

confidence: 55%

mentioning

confidence: 99%

Untangling Planar Curves

Chang

Erickson

2017

Discrete Comput Geom

View full text Add to dashboard Cite

show abstract

“…There are polynomial-time algorithms that reduce any surface graph with 2-terminals [29,65], 3-terminals [33,34,52], and 4-terminals [7,21]. As for arbitrary value of k, previous algorithms assume special positions of the terminals, say when all terminals lie on a single face of the plane graph [18,33].…”

Section: Our Resultsmentioning

confidence: 99%

Tightening Curves on Surfaces Monotonically with Applications

Chang¹,

Mesmay²

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of de Graaf and Schrijver [J. Comb. Theory Ser. B, 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms-the cluster and pipe expansions-to the study of curves on surfaces.As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any k-terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and ∆Y -transformations for arbitrary integer k. Previous algorithms only existed in the planar setting when k ≤ 4, and all of them rely on extensive case-by-case analysis based on different values of k. Our algorithm makes use of the connection between electrical transformations and homotopy moves, and thus solves the problem in a unified fashion.

show abstract

“…transforming a Y-subgraph into a triangle (via center deletion and rewiring). For instance, this property is fulfill by the Peterson graph family forming a Y-∆-equivalence class [3].…”

Section: C4 Y-∆-equivalence Problemmentioning

confidence: 99%

The Graph Grammar Library - A Generic Framework for Chemical Graph Rewrite Systems

Mann

Ekker

Flamm

2013

Theory and Practice of Model Transformations

View full text Add to dashboard Cite

Abstract. Graph rewrite systems are powerful tools to model and study complex problems in various fields of research. Their successful application to chemical reaction modelling on a molecular level was shown but no appropriate and simple system is available at the moment. The presented Graph Grammar Library (GGL) implements a generic Double Push Out approach for general graph rewrite systems. The framework focuses on a high level of modularity as well as high performance, using state-of-the-art algorithms and data structures, and comes with extensive documentation. The large GGL chemistry module enables extensive and detailed studies of chemical systems. It well meets the requirements and abilities envisioned by Yadav et al. (2004) for such chemical rewrite systems. Here, molecules are represented as undirected labeled graphs while chemical reactions are described by according graph grammar rules. Beside the graph transformation, the GGL offers advanced cheminformatics algorithms for instance to estimate energies ofmolecules or aromaticity perception. These features are illustrated using a set of reactions from polyketide chemistry a huge class of natural compounds of medical relevance. The graph grammar based simulation of chemical reactions offered by the GGL is a powerful tool for extensive cheminformatics studies on a molecular level. The GGL already provides rewrite rules for all enzymes listed in the KEGG LIGAND database is freely available at http://www.tbi.univie.ac.at/software/GGL/. BackgroundGraphs are powerful tools to represent and study all kind of data in any field of research. In order to generate graph structures of interest or to alter them according to some directive, graph transformations can be applied. A common approach is to formulate such transformations in terms of graph grammars or graph rewrite systems [5,7,37]. This enables a compact but very expressive representation of allowed alterations and allows for sound mathematical analyses of the problems [2,19].Here, we present our Graph Grammar Library (GGL), a fast and generic C++ framework to formulate and apply graph grammars. Beside the general graph rewrite system, we provide a specialized module to enable efficient graph grammar applications in chemical context. The power of such systems for chemical studies was highlighted by Yadav et al. [49] who emphasized the lack and need for an efficient implementation. The requirements and abilities sketched by Yadav et al. are well met by the capabilites of the GGL framework as presented within this manuscript.To represent molecules in graph notation is well known in chemistry and found in any text book. Therein, atoms are represented by labeled nodes in the graph and the connecting chemical bonds are depicted by according edges (see Fig. 1 a) and b)). Such molecule graphs can be used to model chemical reactions by the application of graph grammar rules as shown in Fig. 1 c) and d). This small example shows already the power of graph rewrite systems, since a single graph grammar rule encodes...

show abstract

Four-terminal reducibility and projective-planar wye-delta-wye-reducible graphs

Cited by 19 publications

References 17 publications

Untangling Planar Curves

Untangling Planar Curves

Tightening Curves on Surfaces Monotonically with Applications

The Graph Grammar Library - A Generic Framework for Chemical Graph Rewrite Systems

Contact Info

Product

Resources

About