A Practical Method for the Minimum Genus of a Graph: Models and Experiments

Beyer, Stephan; Chimani, Markus; Hedtke, Ivo; Kotrbčík, Michal

doi:10.1007/978-3-319-38851-9_6

Cited by 6 publications

(11 citation statements)

References 29 publications

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“…The regions bounded by the edges are the faces of Π. The facial walk model is based on an idea developed in [2] for computing the genus of a graph; it constitutes the only known model for the latter problem. It simulates the face tracing algorithm that visits each face, traversing their borders in clockwise order.…”

Section: Facial Walksmentioning

confidence: 99%

“…An ILP variant where we eliminate the solution variables s e (directly using the containment variables c i a instead) solved 3.29% less instances. We refrain from testing polynomially sized models (betweenness-and index-based instead of constraints (1h-1l)) as our exact genus experiments suggest this does not pay off [2].…”

Section: Algorithm Engineering and Preliminary Benchmarksmentioning

confidence: 99%

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Exact Algorithms for the Maximum Planar Subgraph Problem

Chimani

Hedtke²,

Wiedera

2019

ACM J. Exp. Algorithmics

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Given a graph G, the NP-hard Maximum Planar Subgraph problem asks for a planar subgraph of G with the maximum number of edges. The only known non-trivial exact algorithm utilizes Kuratowski's famous planarity criterion and can be formulated as an integer linear program (ILP) or a pseudo-boolean satisfiability problem (PBS). We examine three alternative characterizations of planarity regarding their applicability to model maximum planar subgraphs. For each, we consider both ILP and PBS variants, investigate diverse formulation aspects, and evaluate their practical performance. ACM Subject Classification Mathematics of computing → Graph algorithms

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Section: Facial Walksmentioning

confidence: 99%

Section: Algorithm Engineering and Preliminary Benchmarksmentioning

confidence: 99%

Exact Algorithms for the Maximum Planar Subgraph Problem

Chimani

Hedtke²,

Wiedera

2019

ACM J. Exp. Algorithmics

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“…For example, the number of rotation systems for the complete graph K 7 is (5!) 7 ≈ 3.6 × 10 14 , and the genus polynomial for K 7 has only recently been computed [2]. Table 1 gives the list of coefficients.…”

Section: Rotation Systemsmentioning

confidence: 99%

“…Each permutation of structure t 1 t 2 2 fixes 8 elements of Σ {0,1,2,3,4} . For instance, (0 2)(1 3) fixes both of the elements with the 1-cycles (0), (2), and (4), both with the 2-cycle (02) and the 1-cycle (4), and also the four elements (0 1)(2 3), (0 3)(1 2), (0 1 2 3), and (0 3 2 1) for a total of 8. The sum of the sized of the fixed-point sets of the four permutations of structure t 1 t 2 2 is 24.…”

Section: Two Roots 2-valent and 3-valentmentioning

confidence: 99%

Calculating genus polynomials via string operations and matrices

et al. 2018

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To calculate the genus polynomials for a recursively specifiable sequence of graphs, the set of cellular imbeddings in oriented surfaces for each of the graphs is usually partitioned into imbedding-types. The effects of a recursively applied graph operation τ on each imbedding-type are represented by a production matrix. When the operation τ amounts to constructing the next member of the sequence by attaching a copy of a fixed graph H to the previous member, Stahl called the resulting sequence of graphs an H-linear family. We demonstrate herein how representing the imbedding types by strings and the operation τ by string operations enables us to automate the calculation of the production matrices, a task requiring time proportional to the square of the number of imbedding-types.

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“…Even for general graphs as small as seven vertices and sparse graphs with around 10 vertices, enumerating topological invariants might require running time in the order of magnitude in hundreds of hours, see [6,9,14,33]. Therefore, our data sets are both challenging and large enough to evaluate the efficiency of the algorithms.…”

Section: Data Setsmentioning

confidence: 99%

Linear Time Canonicalization and Enumeration of Non-Isomorphic 1-Face Embeddings

Hellmuth¹,

Knudsen²,

Kotrbík³

et al. 2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

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Antiparallel strong traces (ASTs) are a type of walks in graphs which use every edge exactly twice. They correspond to 1-face embeddings in orientable surfaces and can be used to design selfassembling protein or DNA strands. Based on a novel canonical form invariant for ASTs, gap vector, we provide a linear-time isomorphism test for ASTs and thus, also for orientable 1-face embeddings of graphs. Using the canonical form, we develop an algorithm for enumerating all pairwise non-isomorphic 1-face embeddings of graphs. We compare our algorithm with an independent implementation of a recent algebraic approach (Bašić et al., MATCH Commun. Math. Comput. Chem. 78 (3), 2017) on large data sets. Our results yield the first large-scale enumeration of nonisomorphic embeddings and investigation of their properties.

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A Practical Method for the Minimum Genus of a Graph: Models and Experiments

Cited by 6 publications

References 29 publications

Exact Algorithms for the Maximum Planar Subgraph Problem

Exact Algorithms for the Maximum Planar Subgraph Problem

Calculating genus polynomials via string operations and matrices

Linear Time Canonicalization and Enumeration of Non-Isomorphic 1-Face Embeddings

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