219 research outputs found
The localization number and metric dimension of graphs of diameter 2
We consider the localization number and metric dimension of certain graphs of diameter , focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with a diameter of , 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 and polarity graphs
The -visibility Localization Game
We study a variant of the Localization game in which the cops have limited
visibility, along with the corresponding optimization parameter, the
-visibility localization number , where is a non-negative
integer. We give bounds on -visibility localization numbers related to
domination, maximum degree, and isoperimetric inequalities. For all , we
give a family of trees with unbounded values. Extending results known
for the localization number, we show that for , every tree contains a
subdivision with . For many , we give the exact value of
for the Cartesian grid graphs, with the remaining cases
being one of two values as long as is sufficiently large. These examples
also illustrate that for all distinct choices of and
$j.
Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings
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 (or,
if the metric has doubling
dimension ), solving a conjecture by Newman and Rabinovich (2020), and
quadratically improving the dependence on the aspect ratio from Indyk et
al.\ (2010). Our second result is a stochastic embedding of a metric space into
trees with expected distortion , 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 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 and recourse 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 recourse, that maintains a matching of weight at
most times the weight of the MST, i.e., a matching of lightness
. We complement our upper bounds with a strategy for an oblivious
adversary that, with recourse , establishes a lower bound of
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
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
A set of isometric paths of a graph is "-rooted", where is a
vertex of , if is one of the end-vertices of all the isometric paths in
. The isometric path complexity of a graph , denoted by , is the
minimum integer such that there exists a vertex satisfying the
following property: the vertices of any isometric path of can be
covered by many -rooted isometric paths.
First, we provide an -time algorithm to compute the isometric path
complexity of a graph with vertices and 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 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
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 and , we consider the
polyhedral complex that parametrizes the directed graph
homomorphisms . 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 for 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
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, -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
In this paper we consider the cop number of graphs with no, or few, short
cycles. We show that when the girth of is at least and the minimum
degree is sufficiently large, where
, then as where . This extends
work of Frankl and implies that if is large and dense in the sense that
while also having girth , then
satisfies Meyniel's conjecture, that is . Moreover, it
implies that if is large and dense in the sense that there for some , while also having girth , then
there exists an such that , thereby
satisfying the weak Meyniel's conjecture. Of course, this implies similar
results for dense graphs with small, that is , numbers of
short cycles, as each cycle can be broken by adding a single cop. We also, show
that there are graphs with girth and minimum degree such that
the cop number is at most . This
resolves a recent conjecture by Bradshaw, Hosseini, Mohar, and Stacho, by
showing that the constant cannot be improved in the exponent of a
lower bound .Comment: 7 pages, 0 figures, 0 table
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