Hasse Diagram Research Articles

Independent coverings and orthogonal colourings

In this paper, two open conjectures are disproved. One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs. A relation between independent coverings and orthogonal colourings is established. This relation is applied to find independent coverings of some sparse partite graphs. Additionally, a degree condition providing the existence of an independent covering in the case where the graph has a square number of vertices is found.

A numeral system for the middle-levels graphs

A sequence S of restricted-growth strings unifies the presentation of middle-levels graphs M k as follows, for 0 < k ∈ Z . Recall M k is the subgraph in the Hasse diagram of the Boolean lattice 2 [2 k +1] induced by the k - and ( k +1)-levels. The dihedral group D 4 k +2 acts on M k via translations mod (2 k + 1) and complemented reversals.The first (2 k )!/( k !( k +1)!) terms of S stand for the orbits of V ( M k ) under such D 4 k +2 -action, via the lexical matching colors 0, 1, ... , k on the k +1 edges at each vertex. So, S is proposed here as a convenient numeral system for the graphs M k . Color 0 allows to reorder S via an integer sequence that behaves as an idempotent permutation on its first (2 k )!/( k !( k +1)!) terms, for each 0 < k ∈ Z . Related properties hold for the remaining colors 1, ... , k .

Fences, their endpoints, and projective Fraïssé theory

We introduce a new class of compact metrizable spaces, which we call fences, and its subclass of smooth fences. We isolate two families F , F 0 \mathcal {F}, \mathcal {F}_{0} of Hasse diagrams of finite partial orders and show that smooth fences are exactly the spaces which are approximated by projective sequences from F 0 \mathcal {F}_{0} . We investigate the combinatorial properties of Hasse diagrams of finite partial orders and show that F , F 0 \mathcal {F}, \mathcal {F}_{0} are projective Fraïssé families with a common projective Fraïssé limit. We study this limit and characterize the smooth fence obtained as its quotient, which we call a Fraïssé fence. We show that the Fraïssé fence is a highly homogeneous space which shares several features with the Lelek fan, and we examine the structure of its spaces of endpoints. Along the way we establish some new facts in projective Fraïssé theory.

Approximation algorithms for maximum weight k-coverings of graphs by packings

Let [Formula: see text] be a family of graphs and let [Formula: see text] be a set of connected graphs, each with at most [Formula: see text] vertices, [Formula: see text] fixed. A [Formula: see text]-packing of a graph GA is a vertex induced subgraph of GA with every connected component isomorphic to a member of [Formula: see text]. A maximum weight [Formula: see text]-covering of a graph GA by [Formula: see text]-packings, is a maximum weight subgraph of GA exactly covered by [Formula: see text] vertex disjoint [Formula: see text]-packings. For a graph [Formula: see text] let [Formula: see text](GA) be a graph, every vertex [Formula: see text] of which corresponds to a vertex subgraph [Formula: see text] of GA isomorphic to a member of [Formula: see text], two vertices [Formula: see text] of [Formula: see text](GA) being adjacent if and only if [Formula: see text] and [Formula: see text] have common vertices or interconnecting edges. The closed neighborhoods containment graph [Formula: see text] of a graph [Formula: see text], is the graph with vertex set [Formula: see text] and edges directed from vertices [Formula: see text] to [Formula: see text] if and only if they are adjacent in GA and the closed neighborhood of [Formula: see text] is contained in the closed neighborhood of [Formula: see text]. A graph [Formula: see text] is a [Formula: see text] reduced graph if it can be obtained from a graph [Formula: see text] by deleting the edges of a transitive subgraph [Formula: see text] of CNCG(GA). We describe 1.582-approximation algorithms for maximum weight [Formula: see text]-coverings by [Formula: see text]-packings of [Formula: see text] and [Formula: see text] reduced graphs when [Formula: see text] is vertex hereditary, has an algorithm for maximum weight independent set and [Formula: see text]. These algorithms can be applied to families of interval filament, subtree filament, weakly chordal, AT-free and circle graphs, to find 1.582 approximate maximum weight [Formula: see text]-coverings by vertex disjoint induced matchings, dissociation sets, forests whose subtrees have at most [Formula: see text] vertices, etc.

Ordering of functions according to multiple fuzzy criteria: application to denoising electroencephalography

We introduce a new relation of order over functions according to multiple fuzzy criteria. Proof of the complied properties for relations of partial orders is given. Convergent and divergent validity of the new membership functions is established. Tolerance to noise of the relation of order is evaluated by corrupting synthetic prototypes and observing changes in the retrieved ordering. The effect of weighting strategies is evaluated in terms of Jaccard and XOR indices. The performance of the ordering algorithm is quantified in terms of richness of the resulting Hasse diagram. Applicability is demonstrated in the context of de-noising electroencephalographic (EEG) signals exemplified over two datasets and evaluated by classification wrapping.

Ptolemaic and Chordal Cover-Incomparability Graphs

Cover-incomparability graphs (C-I graphs) are graphs whose edge-set is the union of edge-sets of the incomparability graph and the cover graph of some poset. C-I graphs captured attention as an interesting class of graphs from posets. It is known that the recognition of C-I graphs is NP-complete (Maxova et al., Order 26(3), 229–236, 2009). Hence, the problem of finding a particular graph family of C-I graphs whose recognition complexity is polynomial is interesting. We present a new forbidden subgraph characterization of Ptolemaic C-I graphs and a linear time algorithm for its recognition. The characterization of chordal C-I graph is an unsolved problem in this area for quite some time. In this paper, we characterize the family of chordal C-I graphs.

(5d RG-flow) trees in the tropical rain forest

5d superconformal field theories (SCFTs) can be obtained from 6d SCFTs by circle compactification and mass deformation. Successive decoupling of hypermultiplet matter and RG-flow generates a decoupling tree of descendant 5d SCFTs. In this paper we determine the magnetic quivers and Hasse diagrams, that encode the Higgs branches of 5d SCFTs, for entire decoupling trees. Central to this undertaking is the approach in [1], which, starting from the generalized toric polygons (GTPs) dual to 5-brane webs/tropical curves, provides a systematic and succinct derivation of magnetic quivers and their Hasse diagrams. The decoupling in the GTP description is straightforward, and generalizes the standard flop transitions of curves in toric polygons. We apply this approach to a large class of 5d KK-theories, and compute the Higgs branches for their descendants. In particular we determine the decoupling tree for all rank 2 5d SCFTs. For each tree, we also identify the flavor symmetry algebras from the magnetic quivers, including non-simply-laced flavor symmetries.

Some inequalities involving the distance signless Laplacian eigenvalues of graphs

‎Given a simple graph $G$‎, ‎the distance signlesss Laplacian‎ ‎$D^{Q}(G)=Tr(G)+D(G)$ is the sum of vertex transmissions matrix‎ ‎$Tr(G)$ and distance matrix $D(G)$‎. ‎In this paper‎, ‎thanks to the‎ ‎symmetry of $D^{Q}(G)$‎, ‎we obtain novel sharp bounds on the distance‎ ‎signless Laplacian eigenvalues of $G$‎, ‎and in particular the‎ ‎distance signless Laplacian spectral radius‎. ‎The bounds are‎ ‎expressed through graph diameter‎, ‎vertex covering number‎, ‎edge‎ ‎covering number‎, ‎clique number‎, ‎independence number‎, ‎domination‎ ‎number as well as extremal transmission degrees‎. ‎The graphs‎ ‎achieving the corresponding bounds are delineated‎. ‎In addition‎, ‎we‎ ‎investigate the distance signless Laplacian spectrum induced by‎ ‎Indu-Bala product‎, ‎Cartesian product as well as extended double‎ ‎cover graph‎.

Attempt to test impact values for multi-indicator systems—exemplified by gender equality

Hasse diagram technique, based on partial ordering is an advantageous method to analyse complex systems of data such as objects characterized by multi-indicator systems. Beside ordered sequences of objects the theory may reveal the impact of each indicator for the set of order relations, which in general is of high interest. A method to measure the impact has already been known for more than 20 years; however, up to now there is no method for a testing as to how far these impacts are considered to be relevant. The actual study presents a first attempt for a testing theory, albeit an extension to statistical testing framework is still open. The testing approach so far is applied to the indicators of gender-equality. It turns out that for the 28 nations (pre-Brexit) of the EU, 2006, in addition to the unified European Union only one indicator can definitely be declared as relevant.

S-fold magnetic quivers

Magnetic quivers and Hasse diagrams for Higgs branches of rank r 4d mathcal{N} = 2 SCFTs arising from ℤℓ mathcal{S} -fold constructions are discussed. The magnetic quivers are derived using three different methods: 1) Using clues like dimension, global symmetry, and the folding parameter ℓ to guess the magnetic quiver. 2) From 6d mathcal{N} = (1, 0) SCFTs as UV completions of 5d marginal theories, and specific FI deformations on their magnetic quiver, which is further folded by ℤℓ. 3) From T-duality of Type IIA brane systems of 6d mathcal{N} = (1, 0) SCFTs and explicit mass deformation of the resulting brane web followed by ℤℓ folding. A choice of the ungauging scheme, either on a long node or on a short node, yields two different moduli spaces related by an orbifold action, thus suggesting a larger set of SCFTs in four dimensions than previously expected.

Classifying Calabi–Yau Threefolds Using Infinite Distance Limits

We present a novel way to classify Calabi–Yau threefolds by systematically studying their infinite volume limits. Each such limit is at infinite distance in Kähler moduli space and can be classified by an associated limiting mixed Hodge structure. We then argue that such structures are labeled by a finite number of degeneration types that combine into a characteristic degeneration pattern associated to the underlying Calabi–Yau threefold. These patterns provide a new invariant way to present crucial information encoded in the intersection numbers of Calabi–Yau threefolds. For each pattern, we also introduce a Hasse diagram with vertices representing each, possibly multi-parameter, decompactification limit and explain how to read off properties of the Calabi–Yau manifold from this graphical representation. In particular, we show how it can be used to count elliptic, K3, and nested fibrations and determine relations of elliptic fibrations under birational equivalence. We exemplify this for hypersurfaces in toric ambient spaces as well as for complete intersections in products of projective spaces.

Ptolemaic and Planar Cover-incomparability Graphs

Cover-Incomparability graphs (C-I graphs) are graphs whose edge-set is the union of edge-sets of incomparability graph and the cover graph of some poset. We give a new characterization of Ptolemaic C-I graphs and prove that each C-I graph contains a Ptolemaic C-I graph as a spanning subgraph. This result is used to present several necessary conditions for a graph G being planar C-I graph. We present a hierarchy of subfamilies of chordal graphs in the class of C-I graphs and prove that many different graph families coincide in the class of C-I graphs.

Hasse diagrams with large chromatic number

For every positive integer n, we construct a Hasse diagram with n vertices and independence number O ( n 3 / 4 ) . Such graphs have chromatic number Ω ( n 1 / 4 ) , which significantly improves the previously best-known constructions of Hasse diagrams having chromatic number Θ ( log n ) . In addition, if we also require girth of at least k ⩾ 5 , we construct such Hasse diagrams with independence number at most n 1 − 1 2 k − 4 + o ( 1 ) . The proofs are based on the existence of point-line arrangements in the plane with many incidences and avoids certain forbidden subconfigurations, which we find of independent interest. These results also have the following surprising geometric consequence. They imply the existence of a family C of n curves in the plane such that the disjointness graph G of C is triangle-free (or has high girth), but the chromatic number of G is polynomial in n. Again, the previously best-known construction, due to Pach, Tardos and Tóth, had only logarithmic chromatic number.

Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.

Non-dominated sorting genetic-based algorithm for exploiting a large-sized fuzzy outranking relation

Electre III is a well-known multiple criteria decision aiding method based on pairwise comparisons. However, it cannot be applied to ranking problems involving many alternatives, because the number of pairwise comparisons can then be rather large. In this paper, we present an evolution-based approach for exploiting large fuzzy outranking relations and deriving a crisp outranking relation with desirable properties. Therefore, the utilization of a fuzzy outranking relation is modeled as a three-objective optimization problem, which is solved by an evolutionary algorithm. The proposed ranking algorithm is a hybrid of the elitist non-dominated sorting genetic algorithm-II (NSGA-II) and a reference point method with the repeated use of a choice mechanism. In addition, a method that portrays the obtained ranking in a Hasse diagram is used for recommendation purposes. We designate the new method RP2-NSGA-II+H. In our experiments, the proposed ranking procedure demonstrates a better performance in terms of ranking error rates than other ranking procedures based on multi-objective evolutionary algorithms. Our experimental results also demonstrate that, with the new procedure, this method can be scaled for hundreds of alternatives.

Analysis of Mine Safety Performance Evaluation Law Based on Matter-Element Analysis and Rough Set of Concept Lattice Reduction

The mine safety performance evaluation and its law were taken as research objects; the human factor, mechanical equipment, management system, environment and technology were taken as mine safety state indexes; accident loss and disabling incident frequency rate as mine safety performance evaluation indexes. The classification standard of safety performance was determined according to the range of influence index and evaluation index, and then the influence index value of safety performance was discretized. According to the classification standard, the classical domain and node domain of safety performance evaluation index classification were determined. Combined with the weight of each safety performance evaluation index, the correlation degree of each mine safety performance evaluation was calculated, and then the mine safety performance was evaluated, and the mine safety performance evaluation decision system was constructed. Based on the dominance-based rough set, the evaluation law of mine safety performance evaluation decision system was analyzed. To generate more reliable laws, an efficient concept lattice reduction method was introduced to reduce the attributes of decision system. The example shows that when the concept lattice is used for reduction, there are 84 concepts in the Hasse diagram, and 24 of them can be identified. The decision system has 8 conditional attributes. And 255 power sets can be obtained under 8 conditional attributes. After eliminating the defect attribute, there are 148 reducible sets, among which there are 4 optimal reducible sets with two conditional attributes. The results show that the proposed method can be used to analyze the law of mine safety performance, so as to improve mine safety work.

Coverings of cubic graphs and 3-edge colorability

Coverings of cubic graphs and 3-edge colorability

Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams

The present paper gives a partial answer to Smale’s question which diagrams can correspond to $(A,B)$-diffeomorphisms. Model diffeomorphisms of the two-dimensional torus derived by “Smale surgery” are considered, and necessary and sufficient conditions for their topological conjugacy are found. Also, a class $G$ of $(A,B)$-diffeomorphisms on surfaces which are the connected sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class $G$ realize any connected Hasse diagrams (abstract Smale graph). Examples of diffeomorphisms from $G$ with isomorphic labeled Smale diagrams which are not ambiently $\Omega$-conjugated are constructed. Moreover, a subset $G_{*}\subset G$ of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient $\Omega$-conjugacy is singled out.

Structure and Hierarchy of SARS-CoV-2 Infection Dynamics Models Revealed by Reaction Network Analysis.

This work provides a mathematical technique for analyzing and comparing infection dynamics models with respect to their potential long-term behavior, resulting in a hierarchy integrating all models. We apply our technique to coupled ordinary and partial differential equation models of SARS-CoV-2 infection dynamics operating on different scales, that is, within a single organism and between several hosts. The structure of a model is assessed by the theory of chemical organizations, not requiring quantitative kinetic information. We present the Hasse diagrams of organizations for the twelve virus models analyzed within this study. For comparing models, each organization is characterized by the types of species it contains. For this, each species is mapped to one out of four types, representing uninfected, infected, immune system, and bacterial species, respectively. Subsequently, we can integrate these results with those of our former work on Influenza-A virus resulting in a single joint hierarchy of 24 models. It appears that the SARS-CoV-2 models are simpler with respect to their long term behavior and thus display a simpler hierarchy with little dependencies compared to the Influenza-A models. Our results can support further development towards more complex SARS-CoV-2 models targeting the higher levels of the hierarchy.

Graph covers with two new eigenvalues

A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve the Sensitivity Conjecture, is closely related to the unique, 4-cycle free, 2-fold cover of the hypercube. We develop a framework in which this connection is a natural first example of the relationship between group valued adjacency matrices with few eigenvalues, and combinatorially interesting covering graphs. In particular, we define a two-eigenvalue cover, to be a covering graph whose adjacency spectra differs (as a multiset) from that of the graph it covers by exactly two eigenvalues. We show that walk regularity of a graph implies walk regularity of any abelian two-eigenvalue cover. We also give a spectral characterization for when a cyclic two-eigenvalue cover of a strongly-regular graph is distance-regular.