219 research outputs found

    The localization number and metric dimension of graphs of diameter 2

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    We consider the localization number and metric dimension of certain graphs of diameter 22, focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with a diameter of 22, we find upper and lower bounds for the localization number and metric dimension, and in many cases these parameters differ only by an additive constant. Our results on the metric dimension of Kneser graphs improve on earlier ones, yielding exact values in infinitely many cases. We determine bounds on the localization number and metric dimension of Moore graphs of diameter 22 and polarity graphs

    The kk-visibility Localization Game

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    We study a variant of the Localization game in which the cops have limited visibility, along with the corresponding optimization parameter, the kk-visibility localization number ζk\zeta_k, where kk is a non-negative integer. We give bounds on kk-visibility localization numbers related to domination, maximum degree, and isoperimetric inequalities. For all kk, we give a family of trees with unbounded ζk\zeta_k values. Extending results known for the localization number, we show that for k2k\geq 2, every tree contains a subdivision with ζk=1\zeta_k = 1. For many nn, we give the exact value of ζk\zeta_k for the n×nn \times n Cartesian grid graphs, with the remaining cases being one of two values as long as nn is sufficiently large. These examples also illustrate that ζiζj\zeta_i \neq \zeta_j for all distinct choices of ii and $j.

    Line Search for an Oblivious Moving Target

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    Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings

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    Low-distortional metric embeddings are a crucial component in the modern algorithmic toolkit. In an online metric embedding, points arrive sequentially and the goal is to embed them into a simple space irrevocably, while minimizing the distortion. Our first result is a deterministic online embedding of a general metric into Euclidean space with distortion O(logn)min{logΦ,n}O(\log n)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\} (or, O(d)min{logΦ,n}O(d)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\} if the metric has doubling dimension dd), solving a conjecture by Newman and Rabinovich (2020), and quadratically improving the dependence on the aspect ratio Φ\Phi from Indyk et al.\ (2010). Our second result is a stochastic embedding of a metric space into trees with expected distortion O(dlogΦ)O(d\cdot \log\Phi), generalizing previous results (Indyk et al.\ (2010), Bartal et al.\ (2020)). Next, we study the \emph{online minimum-weight perfect matching} problem, where a sequence of 2n2n metric points arrive in pairs, and one has to maintain a perfect matching at all times. We allow recourse (as otherwise the order of arrival determines the matching). The goal is to return a perfect matching that approximates the \emph{minimum-weight} perfect matching at all times, while minimizing the recourse. Our third result is a randomized algorithm with competitive ratio O(dlogΦ)O(d\cdot \log \Phi) and recourse O(logΦ)O(\log \Phi) against an oblivious adversary, this result is obtained via our new stochastic online embedding. Our fourth result is a deterministic algorithm against an adaptive adversary, using O(log2n)O(\log^2 n) recourse, that maintains a matching of weight at most O(logn)O(\log n) times the weight of the MST, i.e., a matching of lightness O(logn)O(\log n). We complement our upper bounds with a strategy for an oblivious adversary that, with recourse rr, establishes a lower bound of Ω(lognrlogr)\Omega(\frac{\log n}{r \log r}) for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on Discrete Algorithms (SODA24

    Cyclodextrin Inclusion Complexation for the Efficient Removal of Ochratoxin A from Liquid Food Systems

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    Ochratoxin A (OTA), one major type of mycotoxins, is extensively present in a wide range of food products, such as cereals, fruits, juices, and wine. OTA is the most toxic member of ochratoxins and is classified into Group 2B as a possible human carcinogen by International Agency for Research on Cancer (IARC). To eliminate the OTA contamination in foods, cyclodextrins (CDs) were selected as a promising agent for the OTA removal from the aqueous environment due to its excellent compatibility with OTA, high selectivity, and safety profiles. The supramolecular inclusion complex formation between ochratoxin A and different types of CDs was investigated. Preliminary investigations were carried out on L-Phenylalanine (L-Phe) and four different CDs, including, (2-hydroxypropyl)-α-cyclodextrin (HPαCD), (2-hydroxypropyl)-β-cyclodextrin (HPβCD), heptakis(2,6-di-O-methyl)-β-cyclodextrin (DIMEB), and (2-hydroxypropyl)-γ-cyclodextrin (HPγCD), and results revealed that two β-CD derivatives were more compatible with the benzyl ring on the L-Phe moiety of OTA than HPαCD and HPγCD. Fluorescence studies showed that OTA formed the most stable 1:1 stoichiometric inclusion complex with HPβCD, with a high complexation efficiency of 993.71±51.21 M-1. Together with its safety profiles, HPβCD was determined to be the best OTA-encapsulating candidate. The supramolecular structure, stability, and physicochemical properties of the OTA/HPβCD were then investigated by two-dimensional Rotating-frame Overhauser Effect Spectroscopy (2D 1H-1H ROESY), circular dichroism, quantitative structure-activity relationship (QSAR), and docking analyses. Experimental results and quantum chemical calculations showed clearly that OTA and HPβCD formed a stable guest-host inclusion complex, with the benzyl ring on the L-Phe moiety of OTA (the guest) inserted into the cavity of HPβCD (the host). Thermodynamic studies suggested that OTA/HPβCD complexation was a spontaneous and mainly entropy-driven process, with both steric effect and hydrophobicity of OTA the main driving forces. The efficiency of the complex formation was favoured by low pH, high ionic strength, and the presence of certain simple sugars (glucose, fructose, and sucrose), hydroxy acids (tartaric acid, malic acid, citric acid and lactic acid) and phenolic acids (p-hydroxybenzoic acid, p-coumaric acid, caffeic acid, gallic acid, vanillic acid, and chlorogenic acid); while the process was slightly weakened by the presences of flavanols that would compete with OTA for the binding site of HPβCD. HPβCD was then grafted onto chitosan film which served as a solid support, and the film plasticised with 30% (w/w) of glycerol was found to have the best stability. Fourier transform infrared spectroscopy (FT-IR) and differential scanning calorimetry (DSC) analyses of the film indicated that HPβCD and chitosan interacted via hydrogen bondings and/or Van der Waals interaction and the incorporation of HPβCD increased the thermal stability of the film. The synthesised HPβCD/chitosan film plasticised with 30% (w/w) of glycerol gave the best performance in OTA removal from aqueous systems and commercial grape juice, with the concentration of OTA reduced by 74.10±2.60% and 35.36±2.04%, respectively. Overall, this thesis proved th

    Isometric Path Complexity of Graphs

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    Isometric path complexity of graphs

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    A set SS of isometric paths of a graph GG is "vv-rooted", where vv is a vertex of GG, if vv is one of the end-vertices of all the isometric paths in SS. The isometric path complexity of a graph GG, denoted by ipco(G)ipco(G), is the minimum integer kk such that there exists a vertex vV(G)v\in V(G) satisfying the following property: the vertices of any isometric path PP of GG can be covered by kk many vv-rooted isometric paths. First, we provide an O(n2m)O(n^2 m)-time algorithm to compute the isometric path complexity of a graph with nn vertices and mm edges. Then we show that the isometric path complexity remains bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively studied in Metric Graph Theory. The class of (theta, prism, pyramid)-free graphs are extensively studied in Structural Graph Theory, e.g. in the context of the Strong Perfect Graph Theorem. The class of outerstring graphs is studied in Geometric Graph Theory and Computational Geometry. Our results also show that the distance functions of these (structurally) different graph classes are more similar than previously thought. There is a direct algorithmic consequence of having small isometric path complexity. Specifically, we show that if the isometric path complexity of a graph GG is bounded by a constant, then there exists a polynomial-time constant-factor approximation algorithm for ISOMETRIC PATH COVER, whose objective is to cover all vertices of a graph with a minimum number of isometric paths. This applies to all the above graph classes.Comment: A preliminary version appeared in the proceedings of the MFCS 2023 conferenc

    Homomorphism complexes, reconfiguration, and homotopy for directed graphs

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    The neighborhood complex of a graph was introduced by Lov\'asz to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such `Hom complexes' are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs (digraphs). For any pair of digraphs graphs GG and HH, we consider the polyhedral complex Hom(G,H)\text{Hom}(G,H) that parametrizes the directed graph homomorphisms f:GHf: G \rightarrow H. Hom complexes of digraphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals. We study examples of directed Hom complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a digraph and prove that its homotopy type is recovered as the Hom complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a digraph version of the Mycielski construction, as well as vanishing theorems for higher homology. The Hom complexes of digraphs provide a natural framework for reconfiguration of homomorphisms of digraphs. Inspired by notions of directed graph colorings we study the connectivity of Hom(G,Tn)\text{Hom}(G,T_n) for TnT_n a tournament. Finally, we use paths in the internal hom objects of digraphs to define various notions of homotopy, and discuss connections to the topology of Hom complexes.Comment: 34 pages, 10 figures; V2: some changes in notation, clarified statements and proofs, other corrections and minor revisions incorporating comments from referee

    Geometric Graphs with Unbounded Flip-Width

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    We consider the flip-width of geometric graphs, a notion of graph width recently introduced by Toru\'nczyk. We prove that many different types of geometric graphs have unbounded flip-width. These include interval graphs, permutation graphs, circle graphs, intersection graphs of axis-aligned line segments or axis-aligned unit squares, unit distance graphs, unit disk graphs, visibility graphs of simple polygons, β\beta-skeletons, 4-polytopes, rectangle of influence graphs, and 3d Delaunay triangulations.Comment: 10 pages, 7 figures. To appear at CCCG 202

    Graphs with Large Girth and Small Cop Number

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    In this paper we consider the cop number of graphs with no, or few, short cycles. We show that when the girth of GG is at least 88 and the minimum degree is sufficiently large, δ(lnn)11α\delta \geq (\ln{n})^{\frac{1}{1-\alpha}} where α(0,1)\alpha \in (0,1), then c(G)=o(nδβg4)c(G) = o(n \delta^{\beta -\lfloor \frac{g}{4} \rfloor}) as δ\delta \rightarrow \infty where β>1α\beta> 1-\alpha. This extends work of Frankl and implies that if GG is large and dense in the sense that δn2go(1)\delta \geq n^{\frac{2}{g} - o(1)} while also having girth g8g \geq 8, then GG satisfies Meyniel's conjecture, that is c(G)=O(n)c(G) = O(\sqrt{n}). Moreover, it implies that if GG is large and dense in the sense that there δnϵ\delta \geq n^{\epsilon} for some ϵ>0\epsilon >0, while also having girth g8g \geq 8, then there exists an α>0\alpha>0 such that c(G)=O(n1α)c(G) = O(n^{1-\alpha}), thereby satisfying the weak Meyniel's conjecture. Of course, this implies similar results for dense graphs with small, that is O(n1α)O(n^{1-\alpha}), numbers of short cycles, as each cycle can be broken by adding a single cop. We also, show that there are graphs GG with girth gg and minimum degree δ\delta such that the cop number is at most o(g(δ1)(1+o(1))g4)o(g (\delta-1)^{(1+o(1))\frac{g}{4}}). This resolves a recent conjecture by Bradshaw, Hosseini, Mohar, and Stacho, by showing that the constant 14\frac{1}{4} cannot be improved in the exponent of a lower bound c(G)1g(δ1)g14c(G) \geq \frac{1}{g} (\delta - 1)^{\lfloor \frac{g-1}{4}\rfloor}.Comment: 7 pages, 0 figures, 0 table
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