Graphs, contraction mappings and minors

Weak contraction mapping principle is a generalization of the Banach contraction mapping principle. Weakly contractive mappings are intermediate to contraction mappings and nonexpansive mappings. They have been studied in several contexts. Metric fixed point theory in partially ordered spaces have rapidly developed in recent times. In this paper we extend the concept of weak contraction to subset of a partially ordered generalized metric space which are chains by themselves. It is noted that this weak contraction is different from weak contraction on the whole space. We prove here that under certain assumptions the weakly contractive mapping on certain chains will have a fixed point. Two illustrative examples are given.

Six experiments were conducted to test whether the spatial properties of cognitive maps constructed from verbal descriptions are similar to those of representations derived from visual experience. Data from spatial priming, distance comparison, and mental scanning experiments all suggest that people are able to convert verbal descriptions of configurations into mental representations that reflect the spatial extension of these configurations. In addition, cognitive maps constructed from descriptions exhibit reliable metric properties which make them structurally isomorphic to the corresponding spatial configurations.

The aim of this paper is to study the path connectedness of the minimal and weak minimal solution sets for set optimization problems equipped with partial set order relations defined on the family of bounded sets by means of Minkowski difference.We propose some new notions of generalized convexity for set-valued maps and a lower level map.Further, by investigating several properties of this level map, we study path connectedness of the solution sets for set optimization problems with respect to partial order relations.Finally, we give an application of the derived results to Nash equilibrium games with vector-valued maps under uncertainty.

In this paper, we prove an analogous version of fixed point theorem due to Kannan [Bull. Calcutta Math. Soc., 60, (1968), 71–76] via conditional contractive mappings and some other relational metrical notions. In the proof of present results, we utilize the notion of comparable mappings and some other well known classical fixed point theorems in the settings of relational and ordered metric spaces. We also, highlight the close connection of $$\alpha $$ -admissible mappings with the binary relation and partial order relation as well. Radically, these investigations open another new direction of metric fixed point theory for contractive type mappings. Moreover, non-trivial examples are given to demonstrate the importance and usefulness of such findings.

Science progresses if we succeed in rendering the objects of scientific inquiry more comprehensively or more precisely. Popper tries to formalize this venerable idea. According to him the most comprehensive and most precise description of the world is given by the set T of all true statements. A hypothesis comes the closer to T, or has the more verisimilitude, the more true consequences and the fewer false consequences it implies. Popper proposes to order hypotheses by the inclusion relations between the sets of their true and of their false consequences (“truth contents” and “falsity contents”). A partial ordering would permit one to decide whether the substitution of theory t1 by t2 represents scientific progress. But because of the logical relations between the elements of the sets of logical consequences, or contents, false hypotheses cannot be compared. As our theories usually turn out to be false sooner or later, they can seldom be compared as to their verisimilitude and when they can, the result depends only on which theory implies the other and on their truth-values. Popper even tries to define a measure of verisimilitude on the partial ordering. It has to fail for the same reason. In addition he tries to relativize the concept of a content and fails. What is more, any attempt at defining a measure of better or worse correspondence to the whole truth must fail, as there is no justification for saying that any true primitive sentence asserts as much about reality as some other primitive sentence, or more.

A regressive function (also called a regression or contractive mapping) on a partial order P is a function σ mapping P to itself such that σ(x)≤x. A monotone k-chain for σ is a k-chain on which σ is order-preserving; i.e., a chain x 1<...<xksuch that σ(x 1)≤...≤σ(xk). Let P nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x t(x,j−1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t(е + ek, k) , where ek → 0 as k→∞. Alternatively, the largest k such that every regression on P nis guaranteed to have a monotone k-chain lies between lg*(n) and lg*(n)−2, inclusive, where lg*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.

Almost contractive maps between C⁎-algebras with applications to Fourier algebras

Compatibilité entre structures d'intervalles et relations d'orde

An abstract examination of refinement (and conversely, coarsening) with respect to the involved spatial relations gives rise to formulated order relations between spatial coverings, which are defined as complete-coverage representations composed of regional granules. Coverings, which generalize partitions by allowing granules to overlap, enhance hierarchical geocomputations in several ways. Refinement between spatial coverings has underlying patterns with respectto inclusion—formalized as binary topological relations— between their granules. The patterns are captured by collection relations of inclusion, which are obtained by constraining relevant topological relations with cardinality prop- erties such as uniqueness and totality. Conjoining relevant collection relations of equality and proper inclusion with the overlappedness (non-overlapped or overlapped) of the re- fining and the refined covering yields collection order relations, which serve as specific types of refinement between spatial coverings. The examination results in 75 collection order relations including seven types of equality and 34 pairs of strict or non-strict types of refinement and coarsening, out of which 19 pairs form partial collection orders.

An injective colouring of a graph is a colouring in which every two vertices sharing a common neighbour receive a different colour. Chen, Hahn, Raspaud, and Wang conjectured that every planar graph of maximum degree Δ ≥ 3 admits an injective colouring with at most ⌊3Δ/2⌋ colours. This was later disproved by Lužar and Škrekovski for certain small and even values of Δ and they proposed a new refined conjecture. Using an algorithm for determining the injective chromatic number of a graph, i.e. the smallest number of colours for which the graph admits an injective colouring, we give computational evidence for Lužar and Škrekovski’s conjecture and extend their results by presenting an infinite family of 3-connected planar graphs for each Δ (except for 4) attaining their bound, whereas they only gave finitely many examples for each Δ. Hence, together with another infinite family of maximum degree 4, we provide infinitely many counterexamples to the conjecture by Chen et al. for each Δ if 4 ≤ Δ ≤ 7 and every even Δ ≥ 8. We provide similar evidence for analogous conjectures by La and Štorgel and Lužar, Škrekovski, and Tancer when the girth is restricted as well. Also in these cases we provide infinite families of 3-connected planar graphs attaining the bounds of these conjectures for certain maximum degrees Δ ≥ 3.

Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ Hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput [European J. Combin., 97 (2021), 103395], and that it holds for 4-regular graphs of connectivity 2 with the constant $144 < c$, which we believe to be minimal among all Hamiltonian 4-regular graphs of sufficiently large order. We then disprove a conjecture of Haythorpe by showing that for every nonnegative integer $k$ there is a 5-regular graph on $26 + 6k$ vertices with $2^{k+10} \cdot 3^{k+3}$ Hamiltonian cycles. We prove that for every $d \ge 3$ there is an infinite family of Hamiltonian 3-connected graphs with minimum degree $d$, with a bounded number of Hamiltonian cycles. It is shown that if a 3-regular graph $G$ has a unique longest cycle $C$, at least two components of $G - E(C)$ have an odd number of vertices on $C$, and that there exist 3-regular graphs with exactly two such components.

The aim of this paper is to extend the results of Bhaskar and Lakshmikantham and some other authors and to prove some new coupled fixed point theorems for mappings having a mixed monotone property in a complete metric space endowed with a partial order. Our theorems can be used to investigate a large class of nonlinear problems. As an application, we discuss the existence and uniqueness for a solution of a nonlinear integral equation.

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G+F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k ≤ 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general.In this paper we develop the first algorithm which delivers an optimal solution in polynomial time for every fixed k. In the case when the size of an optimal solution is large compared to k, we also give a min-max formula for the size of a smallest augmenting set.A key step in our proofs is a complete solution of the augmentation problem for a new family of graphs which we call k-independence free graphs. We also prove new splitting off theorems for vertex connectivity.

When a non-numerical ranking is created using Pairwise Comparisons paradigm, its first estimation is a binary relation which may not even be a partial order. In the paper four different partial order approximations of an arbirary binary relation are introduced and discussed.

Big data and the curse of dimensionality are common vocabularies that researchers in different communities have recently been dealing with, e.g. dynamic programming (DP) in automatic control system society. A novel unweighted sampled based least square projection approach is proposed in this study to address the issue of the large state space in the DP optimisation problem. The method, in particular, takes into account both contraction mapping and monotonicity properties of the DP algorithm for value function approximation. Specifically, the batch of samples are gathered by uniform probability distribution at first, and an unweighted LS sub‐problem in the subspace is solved. As the case study, a new Markov decision process model associated with a resource allocation problem is considered to illustrate the technique and evaluate its effectiveness. It is noted that the approach can be employed for different applications as well. Moreover, a MATLAB based software is developed to implement and examine different parts of the proposed method. Simulation examples are considered to support the results of the approach via developed software. The idea makes a connection between the recent advances in big data analysis and approximate DP as well.