The Fixed Vertex Property for Graphs

Analogous to the fixed point property for ordered sets, a graph has the fixed vertex property if each of its endomorphisms has a fixed vertex. The fixed point theory for ordered sets can be embedded into the fixed vertex theory for graphs. Therefore, the potential for cross-fertilization should be explored

Performance guarantees and applications for Xia's algorithm

AbstractXia's algorithm consists of a reduction algorithm and a translation procedure both originally used to tackle the fixed point property for ordered sets. We present results that show that the translation procedure allows access to a much wider range of problems and results showing that the algorithm is very efficient when applied to the fixed point property in ordered sets or for order/graph isomorphism/rigidity

Reconstruction of Finite Truncated Semi-Modular Lattices

AbstractIn this paper, we prove reconstruction results for truncated lattices. The main results are that truncated lattices that contain a 4-crown and truncated semi-modular lattices are reconstructible. Reconstruction of the truncated lattices not covered by this work appears challenging. Indeed, the remaining truncated lattices possess very little lattice-typical structure. This seems to indicate that further progress on the reconstruction of truncated lattices is closely correlated with progress on reconstructing ordered sets in general

A Cacciopoli-Type Inequality to Prove Coercivity of a Bilinear Form Associated with Spatial Hysteresis Internal Damping for an Euler-Bernoulli Beam

We prove an inequality that resembles Cacciopoli inequalities in that it bounds the norm of the derivative of a function by using the norm of the function. Unlike in Cacciopoli inequalities, there is no restriction on the function, a fact made up for by adding an extra term to the norm of the function. The inequality arose in the proof that a bilinear form associated with spatial hysteresis internal damping for an Euler-Bernoulli beam is coercive

The Fixed Point Property in the Set of All Order Relations on a Finite Set

For a finite ground set X, this paper investigates properties of the set of orders with the fixed point property as a subset of the set 𝒪(X) of all orders on X, ordered by inclusion. In particular, it is shown that this set can have singleton components in the covering graph of 𝒪(X), we identify longest possible chains of orders such that orders alternate between having and not having the fixed point property, and we give examples of nondismantlable orders with the fixed point property such that every upper cover in 𝒪(X) has the fixed point property, too

The Use of Retractions in the Fixed Point Theory for Ordered Sets

This chapter gives an overview how retractions are used to prove fixed point results in ordered sets. The primary focus is on comparative retractions, that is, retractions r so that, for each x, the image r(x) is comparable to the preimage x. We start with infinite ordered sets and the classical Abian-Brown Theorem, which establishes that, for every order-preserving self map f of a chain-complete ordered set, if there is an x ≤ f (x), then f has a fixed point. Subsequently, using comparative retractions, we prove that the unit ball in Lp has the (order-theoretical) fixed point property, that is, every order-preserving self map has a fixed point. On a finite ordered set, a comparative retraction is the composition of comparative retractions that each remove a single point. Such a point is called irreducible and the fixed point property is not affected by the presence or absence of irreducible points. An ordered set that can be reduced, by successive removal of irreducible points, to a singleton is called dismantlable by irreducibles. We exhibit the relation between ordered sets that are dismantlable by irreducibles and the application of constraint propagation methods to find fixed point free order-preserving self maps. Closely related to irreducible points are points that are removed by a, not necessarily comparative, retraction that removes a single point. These points are called retractable points. There is a fixed point theorem for retractable points that generalizes the one for irreducible points. However, connectedly collapsible ordered sets, a natural class of ordered sets that is defined based on this theorem, are computationally more challenging than ordered sets that are dismantlable by irreducibles. Whereas the definition of dismantlability can be directly verified in polynomial time, direct verification of the definition of connected collapsibility is worst-case exponential. For graphs, the natural analogue of the fixed point property for ordered sets is the fixed clique property. Although there is no analogue of the Abian-Brown Theorem for the fixed clique property, there are analogues of the fixed point theorems for irreducible and for retractable vertices. For simplicial complexes, the natural analogue of the fixed point property for ordered sets is the fixed simplex property. Although there is an analogue of the fixed point theorem for irreducible vertices, there is no full analogue of the corresponding theorem for retractable vertices. We conclude with the connection between the fixed point property for ordered sets and the iteration of the clique graph operator on the comparability graph. Specifically, it is shown that, for ordered sets that are dismantlable by irreducibles, iteration of the clique graph operator on the comparability graph leads to a graph with one vertex. It is also shown that, if iteration of the clique graph operator on the comparability graph leads to a graph with one vertex, then the ordered set has the fixed point property

The Automorphism Conjecture for Ordered Sets of Dimension 2 and Interval Orders

© 2020, Springer Nature B.V. Let λ ∈ (0, ½). We prove that, for ordered sets P of order dimension 2 and for interval orders, the ratio of the number of automorphisms to the number of endomorphisms is asymptotically bounded by 2−∣P∣λ. The key to the proof is to establish this bound for certain types of lexicographic sums

Minimal Automorphic Superpositions of Crowns

After an overview of uses and explorations of minimal automorphic ordered sets, we present a criterion when certain superpositions of crowns are minimal automorphic. A key lemma to exclude certain retracts can also be applied to ordered sets recently presented in Schröder (Order, https://doi.org/10.1007/s11083-021-09574-3, 2021)

The Fixed Point Property for Ordered Sets of Interval Dimension 2

We provide a polynomial time algorithm that identifies if a given finite ordered set is in the class of d2-collapsible ordered sets. For a d2-collapsible ordered set, the algorithm also determines if the ordered set is connectedly collapsible. Because finite ordered sets of interval dimension 2 are d2-collapsible, in particular, the algorithm determines in polynomial time if a given finite ordered set of interval dimension 2 has the fixed point property. This result is also a first step in investigating the complexity status of the question whether a given collapsible ordered set has the fixed point property

Set Recognition of Decomposable Graphs and Steps Towards Their Reconstruction

It is proved that decomposable graphs are set recognizable and that the index graph of the canonical decomposition as well as the graphs induced on the maximal autonomous sets of vertices are set reconstructible. From these results, we obtain set reconstructibility for many decomposable graphs as well as a concise description of the decomposable graphs for which set reconstruction remains an open problem