Cyclicity of graphs

Hammack, Richard H.

doi:10.1002/(sici)1097-0118(199910)32:2<160::aid-jgt6>3.0.co;2-u

Cited by 10 publications

(5 citation statements)

References 2 publications

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“…It is possible that some similar properties hold for k > 1 since the key condition in the recursion, that two cycles being added meet in a common nontrivial path, generalizes -the connected sum of two k-spheres remains a k-sphere. But B, B ′ aren't panaceas [18].…”

Section: Some Properties Of the Basesmentioning

confidence: 99%

Canonical sphere bases for simplicial and cubical complexes

Kainen

2023

J. Knot Theory Ramifications

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Sphere-bases are constructed for the [Formula: see text] vector space formed by the [Formula: see text]-dimensional subcomplexes, of [Formula: see text]-simplex (or [Formula: see text]-cube), for which every [Formula: see text]-face is contained in a positive even number of [Formula: see text]-cells; addition is symmetric difference of the corresponding sets of [Formula: see text]-cells. The bases consist of the boundaries of an algorithmically-specified family of [Formula: see text]-simplexes or [Formula: see text]-cubes. Properties of the bases are investigated.

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Section: Some Properties Of the Basesmentioning

confidence: 99%

Canonical sphere bases for simplicial and cubical complexes

Kainen

2023

J. Knot Theory Ramifications

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“…A general method for generating cycle bases for Cartesian product graphs, was described by Hammack [6]. It constructs a cycle basis for G × G for each pair of graphs G, G with given spanning trees T, T and cycle bases B, B for G, G , resp.…”

Section: Remarksmentioning

confidence: 99%

Cycle construction and geodesic cycles with application to the hypercube

Kainen

2014

Ars Math. Contemp.

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Construction of cycles in a graph is investigated, where cycles from particular subsets (such as bases) are added together so that each partial sum is also a cycle or each new cycle intersects the sum of the preceding terms in a nontrivial path. Starting with the geodesic cycles, a hierarchical construction is given. For the hypercube graph, geodesic cycles are characterized, and it is shown how hypercube geodesic cycles can be constructed in two steps from a special basis. Applications are given to inferring commutativity of a diagram in a groupoid from commutativity of some of its cycles.

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“…Of particular relevance is the closely related problem of determining the cyclicity [10] of a graph, that is, the length of a longest cycle to which a given graph can be contracted. Cyclicity was introduced by Blum [3] under the name co-circularity, due to a close relationship with a concept in topology called circularity (see also [1]).…”

Section: Introductionmentioning

confidence: 99%

“…Cyclicity was introduced by Blum [3] under the name co-circularity, due to a close relationship with a concept in topology called circularity (see also [1]). Later Hammack [10] coined the current name for the concept and gave both structural results and polynomial-time algorithms for a number of special graph classes. He also proved that the problem of determining the cyclicity is NP-hard for general graphs [11].…”

Section: Introductionmentioning

confidence: 99%

“…Hammack [10] proved that C k -Contractibility is polynomial-time solvable on planar graphs for every k ≥ 3. Later, Kamiński, Paulusma and Thilikos [15] proved that H-Contractibility is polynomial-time solvable on planar graphs for every graph H. Golovach, Kratsch and Paulusma [9] proved that the H-Contractibility problem is polynomial-time solvable on AT-free graphs for every triangle-free graph H. Hence, as C 3 -Contractibility is readily seen to be polynomial-time solvable for general graphs, C k -Contractibility and P k -Contractibility are polynomial-time solvable on AT-free graphs for every integer k ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

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Contracting Bipartite Graphs to Paths and Cycles

Dabrowski

Paulusma

2017

Electronic Notes in Discrete Mathematics

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Testing if a given graph G contains the k-vertex path P k as a minor or as an induced minor is trivial for every fixed integer k ≥ 1. However, the situation changes for the problem of checking if a graph can be modified into P k by using only edge contractions. In this case the problem is known to be NP-complete even if k = 4. This led to an intensive investigation for testing contractibility on restricted graph classes. We focus on bipartite graphs. Heggernes, van 't Hof, Lévêque and Paul proved that the problem stays NPcomplete for bipartite graphs if k = 6. We strengthen their result from k = 6 to k = 5. We also show that the problem of contracting a bipartite graph to the 6-vertex cycle C6 is NP-complete. The cyclicity of a graph is the length of the longest cycle the graph can be contracted to. As a consequence of our second result, determining the cyclicity of a bipartite graph is NP-hard.

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Cyclicity of graphs

Cited by 10 publications

References 2 publications

Canonical sphere bases for simplicial and cubical complexes

Canonical sphere bases for simplicial and cubical complexes

Cycle construction and geodesic cycles with application to the hypercube

Contracting Bipartite Graphs to Paths and Cycles

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