Graphs and Cubes

Овчинников, С. Г.

doi:10.1007/978-1-4614-0797-3

Cited by 21 publications

(19 citation statements)

References 59 publications

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“…The reader is referred to the books of Ovchinnikov [16] and Imrich and Klavžar [14] for known results on partial cubes. Let us also mention that some other structures previously defined in the literature are essentially equivalent to partial cubes.…”

Section: Classes Of Partial Cubesmentioning

confidence: 99%

Covering Partial Cubes with Zones

Cardinal

Felsner

2014

Lecture Notes in Computer Science

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Abstract. A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite poset. For several such special cases, we give upper and lower bounds on the minimum size of a covering by zones. We also consider the computational complexity of those problems, and establish some hardness results.

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Section: Classes Of Partial Cubesmentioning

confidence: 99%

Covering Partial Cubes with Zones

Cardinal

Felsner

2014

Lecture Notes in Computer Science

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“…It is well-known that partial cubes have a unique embedding in Q up to automorphisms of Q , see e.g. [32,Chapter 5]. In other words, the tope graph of a simple partial cube system is invariant under reorientation.…”

Section: Systems Of Sign-vectors and Partial Cubesmentioning

confidence: 99%

On Tope Graphs of Complexes of Oriented Matroids

Knauer

Marc

2019

Discrete Comput Geom

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We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs.Further corollaries include a characterization of topes of oriented matroids due to da Silva, another one of Handa, a characterization of lopsided systems due to Lawrence, and an intrinsic characterization of tope graphs of affine oriented matroids. Moreover, we obtain purely graph theoretic polynomial time recognition algorithms for tope graphs of the above and a finite list of excluded partial cube minors for the bounded rank case.In particular, our results answer a relatively long-standing open question in oriented matroids and can be seen as identifying the theory of (complexes of) oriented matroids as a part of metric graph theory. Another consequence is that all finite Pasch graphs are tope graphs of complexes of oriented matroids, which confirms a conjecture of Chepoi and the two authors.

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“…This of course implies that the isomorphism of (explicitly given) partial cubes is already GI-complete as median graphs are partial cubes (see e.g. [17], Theorem 5.75).…”

Section: Lemmamentioning

confidence: 99%

“…. What we call a signed permutation of Hn, Ovchinnikov[17] (Section 3.8) calls an automorphism of the cube H(X) of a set X: the automorphism group of H(X) (which is the same as Hn if |X| = n) is the semidirect product of all permutations of X and the elementary Abelian 2-group on X. In other words, the automorphism group of H(X) consists of all signed permutations on |X| variables.…”

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confidence: 99%

Solution-Graphs of Boolean Formulas and Isomorphism1

Scharpfenecker

Torán

2019

SAT

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The solution-graph of a Boolean formula on n variables is the subgraph of the hypercube H n induced by the satisfying assignments of the formula. The structure of solution-graphs has been the object of much research in recent years since it is important for the performance of SAT-solving procedures based on local search. Several authors have studied connectivity problems in such graphs focusing on how the structure of the original formula might affect the complexity of the connectivity problems in the solution-graph. In this paper we study the complexity of the isomorphism problem of solution-graphs of Boolean formulas. We consider the classes of formulas that arise in the CSP-setting and investigate how the complexity of the isomorphism problem depends on the formula type. We observe that for general formulas the solution-graph isomorphism problem can be solved in exponential time while in the cases of 2CNF formulas, as well as for CPSS formulas, the problem is in the counting complexity class C = P, a subclass of PSPACE. We also prove a strong property on the structure of solution-graphs of Horn formulas showing that they are just unions of partial cubes. In addition, we give a PSPACE lower bound for the problem on general Boolean functions. We prove that for 2CNF, as well as for CPSS formulas the solution-graph isomorphism problem is hard for C = P under polynomial time many-one reductions, thus matching the given upper bound.

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Graphs and Cubes

Cited by 21 publications

References 59 publications

Covering Partial Cubes with Zones

Covering Partial Cubes with Zones

On Tope Graphs of Complexes of Oriented Matroids

Solution-Graphs of Boolean Formulas and Isomorphism1

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