Enumerating the Walecki-Type Hamiltonian Cycle Systems

Brugnoli, Emanuele

doi:10.1002/jcd.21558

Cited by 2 publications

(2 citation statements)

References 16 publications

(20 reference statements)

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“…Kelly's conjecture has been verified for regular tournaments on n vertices, whenever n is sufficiently E-mail address: janez.ales@gmail.com (Janez Aleš) c b This work is licensed under http://creativecommons.org/licenses/by/3.0/ large in [8]. Counting Walecki-type Hamiltonian cycle systems up to isomorphism has been solved by Brugnoli [6]. The problem of enumerating non-isomorphic Walecki tournaments has not been solved to date.…”

Section: Introductionmentioning

confidence: 99%

Automorphism groups of Walecki tournaments with zero and odd signatures

Aleš

2018

Art Discrete Appl. Math.

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Walecki tournaments were defined by Alspach in 1966. They form a class of regular tournaments that posses a natural Hamilton directed cycle decomposition. It has been conjectured by Kelly in 1964 that every regular tournament possesses such a decomposition. Therefore Walecki tournaments speak in favor of the conjecture. A second interest in Walecki tournaments arises from the mapping between cycles of the complementing circular shift register and isomorphism classes of Walecki tournaments. The problem of enumerating non-isomorphic Walecki tournaments has not been solved to date. We characterize the arc structure of Walecki tournaments whose corresponding binary sequences have zero and odd signature. Automorphism groups are determined for zero signature Walecki tournaments and for odd signature Walecki tournaments with the zero signature Walecki subtournaments. Walecki tournaments possess a broad range of subtournaments isomorphic to some Walecki tournament. Subtournaments of odd signature Walecki tournaments induced by the outsets of the central vertex are proven to be either regular or almost regular.

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Section: Introductionmentioning

confidence: 99%

Automorphism groups of Walecki tournaments with zero and odd signatures

Aleš

2018

Art Discrete Appl. Math.

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“…In Chapter 6 we will show that 1-singular DP-reduction for 2-CNF MUs is closely related to the homeomorphism of their implication graphs. Furthermore we obtain full understanding of these homeomorphism types (and so normalforms), namely that they correspond to the class of binary strings called "bracelets", or "turnover necklaces" ( [61], [25]; see Definition 6.5.8).…”

Section: The Program Of "Classifying Mus"mentioning

confidence: 99%

The combinatorics of minimal unsatisfiability: connecting to graph theory

Abbasizanjani¹

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Minimally Unsatisﬁable CNFs (MUs) are unsatisﬁable CNFs where removing any clause destroys unsatisﬁability. MUs are the building blocks of unsatisﬁa-bility, and our understanding of them can be very helpful in answering various algorithmic and structural questions relating to unsatisﬁability. In this thesis we study MUs from a combinatorial point of view, with the aim of extending the understanding of the structure of MUs. We show that some important classes of MUs are very closely related to known classes of digraphs, and using arguments from logic and graph theory we characterise these MUs.Two main concepts in this thesis are isomorphism of CNFs and the implica-tion digraph of 2-CNFs (at most two literals per disjunction). Isomorphism of CNFs involves renaming the variables, and ﬂipping the literals. The implication digraph of a 2-CNF F has both arcs (¬a → b) and (¬b → a) for every binary clause (a ∨ b) in F .In the ﬁrst part we introduce a novel connection between MUs and Minimal Strong Digraphs (MSDs), strongly connected digraphs, where removing any arc destroys the strong connectedness. We introduce the new class DFM of special MUs, which are in close correspondence to MSDs. The known relation between 2-CNFs and implication digraphs is used, but in a simpler and more direct way, namely that we have a canonical choice of one of the two arcs. As an application of this new framework we provide short and intuitive new proofs for two im-portant but isolated characterisations for nonsingular MUs (every literal occurs at least twice), both with ingenious but complicated proofs: Characterising 2-MUs (minimally unsatisﬁable 2-CNFs), and characterising MUs with deﬁciency 2 (two more clauses than variables).In the second part, we provide a fundamental addition to the study of 2-CNFs which have eﬃcient algorithms for many interesting problems, namely that we provide a full classiﬁcation of 2-MUs and a polytime isomorphism de-cision of this class. We show that implication digraphs of 2-MUs are “Weak Double Cycles” (WDCs), big cycles of small cycles (with possible overlaps). Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry (a self-inverse ﬁxed-point free anti-symmetry, re-versing the direction of arcs). It follows that the isomorphisms between 2-MUs are exactly the isomorphisms between their implication digraphs (since digraphs with given skew-symmetry are the same as 2-CNFs). This reduces the classiﬁ-cation of 2-MUs to the classiﬁcation of a nice class of digraphs.Finally in the outlook we discuss further applications, including an alter-native framework for enumerating some special Minimally Unsatisﬁable Sub-clause-sets (MUSs).

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Enumerating the Walecki-Type Hamiltonian Cycle Systems

Cited by 2 publications

References 16 publications

Automorphism groups of Walecki tournaments with zero and odd signatures

Automorphism groups of Walecki tournaments with zero and odd signatures

The combinatorics of minimal unsatisfiability: connecting to graph theory

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