Unavoidable Set Research Articles

Automated testing and interactive construction of unavoidable sets for graph classes of small path‐width

AbstractLet be a class of graphs with a membership test, , and let be the class of graphs in of path‐width at most . We present an interactive framework that finds an unavoidable set for , which is a set of graphs such that any graph in contains an isomorphic copy of a graph in . At the core of our framework is an algorithm that verifies whether a set of graphs is, indeed, unavoidable for . While obstruction sets are well‐studied, so far there is no general theory or algorithm for finding unavoidable sets. In general, it is undecidable whether a finite set of graphs is unavoidable for a given graph class. However, we give a criterion for termination: our algorithm terminates whenever is locally checkable of bounded maximum degree and is a finite set of connected graphs. For example, ‐regular graphs, ‐colourable graphs, and ‐free graphs are locally checkable classes. We put special emphasis on the case that is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high‐degree‐first path‐decompositions, which enables highly efficient pruning techniques. We exploit our framework to prove a new lower bound on the path‐width of cubic graphs. Moreover, we determine the extremal girth values of cubic graphs of path‐width for all and all smallest graphs which take on these extremal girth values. Further, we present a new constructive characterisation of the extremal cubic graphs of path‐width 3 and girth 4.

Divisibility of integers obtained from truncated periodic sequences

A finite set of prime numbers [Formula: see text] is called unavoidable with respect to [Formula: see text] if for each [Formula: see text] the sequence of integer parts [Formula: see text], [Formula: see text] contains infinitely many elements divisible by at least one prime number [Formula: see text] from the set [Formula: see text]. It is known that an unavoidable set exists with respect to [Formula: see text] and that it does not exist if [Formula: see text] is an integer such that [Formula: see text] is not square free. In this paper, we show that no finite unavoidable sets exist with respect to [Formula: see text] if [Formula: see text] is a prime number or [Formula: see text] belongs to some explicitly given arithmetic progressions, for instance, [Formula: see text] and [Formula: see text], [Formula: see text]

The Reductivity of Spherical Curves Part II: 4-gons

The reductivity of a spherical curve represents how reduced it is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four, that is, we give a set of parts of spherical curves such that every spherical curve with reductivity four has at least one of the parts, by considering 4-gons.

Neutron interferometer crystallographic imperfections and gravitationally induced quantum interference measurements

Dynamical diffraction leads to an interesting, unavoidable set of interference effects for neutron interferometers. This experiment studies the interference signal from two and three successive Bragg diffractions in the Laue geometry. We find that intrinsic Bragg-plane misalignment in monolithic, "perfect" silicon neutron interferometers is relevant between successive diffracting crystals, as well as within the Borrmann fan for typical interferometer geometries. We show that the dynamical phase correction employed in the Colella, Overhauser, and Werner gravitationally induced quantum interference experiments is attenuated by slight, intrinsic misalignments between diffracting crystals, potentially explaining the long-standing 1% discrepancy between theory and experiment. This systematic may also impact precision measurements of the silicon structure factor, affecting previous and future measurements of the Debye-Waller factor and neutron-electron scattering length as well as potential fifth-force searches. For the interferometers used in this experiment, Bragg planes of different diffracting crystals were found to be misaligned by 10 to 40 nrad.

Optimal unavoidable sets of types of 3-paths for planar graphs of given girth

In this paper we study unavoidable sets of types of 3-paths for families of planar graphs with minimum degree at least 2 and a given girth g. A 3-path of type (i,j,k) is a path uvw on three vertices u, v, and w such that the degree of u (resp. v, resp. w) is at most i (resp. j, resp. k). The elements i,j,k are called parameters of the type. The set S of types of paths is unavoidable for a family F of graphs if each graph G from F contains a path of the type from S. An unavoidable set S of types of paths is optimal for the family F if neither any type can be omitted from S, nor any parameter of any type from S can be decreased.We prove that the set Sg (resp. S′g) is an optimal set of types of 3-paths for the family of plane graphs having δ(G)≥2 and girth g(G)≥g where (i)S5={(2,∞,2),(2,3,5),(2,4,3),(3,3,3)},(ii)S7={(2,3,3),(2,5,2)},S7′={(2,2,6),(2,3,3),(2,4,2)},(iii)S8={(2,2,5),(2,3,2)},(iv)S10={(2,4,2)},S10′={(2,2,3),(2,3,2)},(v)S11={(2,2,3)}.

Ambiguity in the heart, art and science by Marcus C. Schaub

Fig. 1. Ikaros in the Sky. Cutout colored paper in 1943 by Henri Matisse. Published in “Jazz” by Teriade, Edition Verve, Paris, 1947. 7th International AsconaWorkshop on Cardiomyocyte Biology, which focused on cardiac pathways of differentiation, metabolism and contraction, reveals a common notion appearing towards the end in each of them. Most material deals with basic science, but opens by mentioning the demographic socio-economic burden of high morbidity and mortality of cardiovascular diseases in technologically developed countries. Usually the circle gets closed at the end of the article. The endings are characterized by a transition from the precisely defined argumentative presentation of scientific facts to a more open not to say vague escape into speculation on “... potential opening of avenues for new therapeutic strategies”. We all know the trick that is needed to land research grant support. However, such ambiguity is not only reigning in science but is also typical for the representational arts. For illustration let's start with a paradigmatic aside by viewing Henri Matisse's picture of Ikaros in the Sky (Fig. 1). The dominant feature in this picture is his radiant red heart. It may serve as a metaphor for the ambiguity in art and science. We all know the story of Daidalos, the doyen of the Athenian craftsmanship, and his son Ikaros,whowere kept captive onKreta by the king Minos. To fly from the island Kreta, Daidalos constructed wings for his son and himself, by fixing together bird's feathers with wax. The father warned his son not to fly too high getting too close to the sun; otherwise, the sun's heat would melt the wax, loosen the feathers and cause him to fall back to earth. Ikaros, however, enjoying the newly gained freedom of flight couldn't help flying higher and higher into the sky until the unavoidable fate set in and he lost his feathers. The French painter Henri Matisse just captured this very moment of Ikaros using cutouts of colored paper (Fig. 1) when he had to give up brush painting because of severe polyarthritis. What is Matisse showing in this cutout? Has Ikaros resigned and is falling back, or is he still struggling to gain height? He just might experience an unprecedented new feeling of elation at the very moment of absolute weightlessness before the spacecraft era. Whatever he feels, he seems to feel it by heart. Ikaros has his heart on the right spot, which is onhis left;we, however, see it on the right. This is unlikely to be amedical problem; Ikaroswas not reported to suffer fromtheKartagener syndrome with “situs inversus” (described in 1933 by Max Kartagener, a physician fromZurich). It'smore likely to be an optical gimmick.Weperceive Ikaros as amirror image as if we looked ourselves into amirror; andwhat is the mirror doing? It simply reverses front and back (but neither left–right nor up–down). This reveals another ambiguity inMatisse's cutout; are we

Unavoidable topological minors of infinite graphs

A graph G is loosely- c -connected, or ℓ - c -connected, if there exists a number d depending on G such that the deletion of fewer than c vertices from G leaves precisely one infinite component and a graph containing at most d vertices. In this paper, we give the structure of a set of ℓ - c -connected infinite graphs that form an unavoidable set among the topological minors of ℓ - c -connected infinite graphs. Corresponding results for minors and parallel minors are also obtained.

An unavoidable set of $D$-reducible configurations

We give a new proof of the four-color theorem by exhibiting an unavoidable set of 2822 D-reducible configurations. The existence of such a set had been conjectured by several researchers including Stromquist (1975), Appel and Haken (1977), and Robertson, Sanders, Seymour and Thomas (1997).

Unavoidable Sets of Partial Words

The notion of an unavoidable set of words appears frequently in the fields of mathematics and theoretical computer science, in particular with its connection to the study of combinatorics on words. The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words. Partial words, or finite sequences that may contain a number of “do not know” symbols or “holes,” appear naturally in several areas of current interest such as molecular biology, data communication, and DNA computing. We demonstrate the utility of the notion of unavoidability of sets of partial words by making use of it to identify several new classes of unavoidable sets of full words. Along the way we begin work on classifying the unavoidable sets of partial words of small cardinality. We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two. We give a result which makes the conjecture easy to verify for a significant number of cases. We characterize many forms of unavoidable sets of partial words of size three over a binary alphabet, and completely characterize such sets over a ternary alphabet. Finally, we extend our results to unavoidable sets of partial words of size k over a k-letter alphabet.

Unavoidable sets

We obtain new results on minimum lengths of words in an unavoidable set of words of cardinality n before introducing the notion of aperiodic unavoidable sets, a natural extension of unavoidable sets.

UNAVOIDABLE AND ALMOST UNAVOIDABLE SETS OF WORDS

A set of words over a finite alphabet is called an unavoidable set if every word of sufficiently long length must contain some word from this set as a subword. Motivated by a theorem from automata theory, we introduce the notion of an almost unavoidable set and prove certain asymptotic estimates for the size of almost unavoidable sets of uniform length.

UNAVOIDABLE SETS OF CONSTANT LENGTH

A set of words X is called unavoidable on a given alphabet A if every infinite word on A has a factor in X. For k,q≥1, let c(k,q) be the number of conjugacy classes of words of length k on q letters. An unavoidable set of words of length k on q symbols has at least c(k,q) elements. We show that for any k,q≥1, there exists an unavoidable set of words of length k on q symbols having c(k,q) elements.

Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. II: Concave operators

We describe the set of propagation of maxima of upper semicontinuous viscosity subsolutions of fully nonlinear, degenerate elliptic Hamilton-Jacobi-Bellman equations in the concave case, i.e., for operators represented as the minimum of a parametrized family of 2 nd order linear operators. We show that the set where an interior maximum propagates contains the reachable set of a suitable deterministic differential game, as well as the unavoidable set of a controlled diffusion process. We also obtain a Strong Maximum Principle for a class of operators that are not strictly elliptic.

Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. I: Convex operators

We describe the set of propagation of maxima of upper semicontinuous viscosity subsolutions of fully nonlinear, degenerate elliptic Hamilton-Jacobi-Bellman equations in the con- cave case, i.e., for operators represented as the minimum of a parametrized family of 2 nd order linear operators. We show that the set where an interior maximum propagates contains the reach- able set of a suitable deterministic diVerential game, as well as the unavoidable set of a controlled diVusion process. We also obtain a Strong Maximum Principle for a class of operators that are not strictly elliptic.

Unavoidable Set: Extension and Reduction

We give an explicit criterion for unavoidability of word sets. We characterize extendible, finitely and infinitely as well, elements in them. We furnish a reasonable upper bound and an exponential lower bound on the maximum leghth of words in a reduced unavoidable set of a given cardinality.

Unavoidable set of face types for planar maps

Unavoidable set of face types for planar maps

Using unavoidable set of trees to generalize Kruskal's theorem

Presentation d'une application simple, consistant a demontrer la terminaison d'un systeme de reecriture particulier, compose d'une regle unique

On extendibility of unavoidable sets

A subset X of a free monoid A ∗ is said to be unavoidable if all but finitely many words in A ∗ contain some word of X as a subword. A. Ehrenfeucht has conjectured that every unavoidable set X is extendible in the sense that there exist x ϵ X and a ϵ A such that ( X − { x }) ∪ { xa } is itself unavoidable. This problem remains open, we give some partial solutions and show how to efficiently test unavoidability, extendibility and other properties of X related to the problem.

An optimal test on finite unavoidable sets of words

We give an optimal bound on the maximal length of a word that avoids a finite unavoidable set of words.

The Nature of Proof: Limits and Opportunities

The fundamental reason that the unavoidable set argument worked whereas other approaches to the Four-Color Conjecture did not is that all other approaches need somewhat stronger theoretical tools to make their methods apply. While these might be possible to create, there is no guarantee that they are actually possible; and if they are, there is no obvious way to go about finding them. On the other hand, many mathematicians have believed that an unavoidable set of reducible configurations might exist, but that a smallest such set was beyond the bounds of reasonable computation. This attitude appears justified when the problem is considered with respect to the tools available prior to 1960. After 1960, with the advent of faster computers, there were still strong reasons to believe that the computations would be infeasibly large, but there were certainly no theoretical difficulties to overcome other than the choice of a method for obtaining an unavoidable set. Thus by 1970 it became a problem of discovering whether efficient use of known techniques and technical (as opposed to theoretical) improvements would enable one to find an unavoidable set of reducible configurations. Most mathematicians who were educated prior to the development of fast computers tend not to think of the computer as a routine tool to be used in conjunction with other older and more theoretical tools in advancing mathematical knowledge. Thus they intuitively feel that if an argument contains parts that are not verifiable by hand calculations it is on rather insecure ground. There is a tendency to feel that verification of computer results by independent computer programs is not as certain to be correct as independent hand checking of the proof of theorems proved in the standard way.