scholarly journals The minimum restricted edge-connected graph and the minimum size of graphs with a given edge–degree

2014 ◽  
Vol 167 ◽  
pp. 304-309
Author(s):  
Weihua Yang ◽  
Yingzhi Tian ◽  
Hengzhe Li ◽  
Hao Li ◽  
Xiaofeng Guo
Keyword(s):  
Connected Graph ◽  
Minimum Size

scholarly journals The Cost of 2-Distinguishing Cartesian Powers

10.37236/3223 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Debra Boutin
Keyword(s):  
Connected Graph ◽  
Minimum Size ◽  
Cartesian Power ◽  
The Cost ◽  
Pointwise Stabilizer

A graph $G$ is said to be $2$-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the label classes.  The minimum size of a label class in any such labeling of $G$ is called the cost of $2$-distinguishing $G$ and is denoted by $\rho(G)$.  The determining number of a graph $G$, denoted $\det(G)$, is the minimum size of a set of vertices whose pointwise stabilizer is trivial.  The main result of this paper is that if $G^k$ is a $2$-distinguishable Cartesian power of a prime, connected graph $G$ on at least three vertices with $\det(G)\leq k$ and $\max\{2, \det(G)\} < \det(G^k)$, then $\rho(G^k) \in \{\det(G^k), \det(G^k)+1\}$.  In particular, for $n\geq 3$, $\rho(K_3^n)\in \{ \left\lceil {\log_3 (2n+1)} \right\rceil$ $+1, \left\lceil {\log_3 (2n+1)} \right\rceil$ $+2\}$.

On the Steiner 4-Diameter of Graphs

2018 ◽  
Vol 18 (01) ◽  
pp. 1850002 ◽  
Author(s):  
ZHAO WANG ◽  
YAPING MAO ◽  
HENGZHE LI ◽  
CHENGFU YE
Keyword(s):  
Connected Graph ◽  
Natural Generalization ◽  
Minimum Size ◽  
Graph Distance ◽  
Connected Subgraphs ◽  
Vertex Sets

The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph G of order at least 2 and [Formula: see text], the Steiner distance dG(S) among the vertices of S is the minimum size among all connected subgraphs whose vertex sets contain S. Let n, k be two integers with 2 ≤ k ≤ n. Then the Steiner k-eccentricity ek(v) of a vertex v of G is defined by [Formula: see text]. Furthermore, the Steiner k-diameter of G is [Formula: see text]. In 2011, Chartrand, Okamoto and Zhang showed that k − 1 ≤ sdiamk(G) ≤ n − 1. In this paper, graphs with sdiam4(G) = 3, 4, n − 1 are characterized, respectively.

scholarly journals On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

2021 ◽  
Author(s):  
Waldo Gálvez ◽  
Fabrizio Grandoni ◽  
Afrouz Jabal Ameli ◽  
Krzysztof Sornat
Keyword(s):  
Approximation Algorithms ◽  
Connected Graph ◽  
Minimum Size ◽  
Single Cycle ◽  
Integrality Gap ◽  
Connectivity Augmentation ◽  
Special Case ◽  
Better Than

AbstractIn the k-Connectivity Augmentation Problem we are given a k-edge-connected graph and a set of additional edges called links. Our goal is to find a set of links of minimum size whose addition to the graph makes it (k + 1)-edge-connected. There is an approximation preserving reduction from the mentioned problem to the case k = 1 (a.k.a. the Tree Augmentation Problem or TAP) or k = 2 (a.k.a. the Cactus Augmentation Problem or CacAP). While several better-than-2 approximation algorithms are known for TAP, for CacAP only recently this barrier was breached (hence for k-Connectivity Augmentation in general). As a first step towards better approximation algorithms for CacAP, we consider the special case where the input cactus consists of a single cycle, the Cycle Augmentation Problem (CycAP). This apparently simple special case retains part of the hardness of the general case. In particular, we are able to show that it is APX-hard. In this paper we present a combinatorial $\left (\frac {3}{2}+\varepsilon \right )$ 3 2 + ε -approximation for CycAP, for any constant ε > 0. We also present an LP formulation with a matching integrality gap: this might be useful to address the general case of the problem.

scholarly journals Bounds for Identifying Codes in Terms of Degree Parameters

10.37236/2036 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Florent Foucaud ◽  
Guillem Perarnau
Keyword(s):  
Large Class ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Clique Number ◽  
Minimum Size ◽  
Regular Graphs ◽  
Identifying Codes ◽  
Identifying Code ◽  
Sharp Estimates

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\gamma^{\text{ID}}(G)$ denotes the minimum size of an identifying code of a graph $G$, it was conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there exists a constant $c$ such that if a connected graph $G$ with $n$ vertices and maximum degree $d$ admits an identifying code, then $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{d}+c$. We use probabilistic tools to show that for any $d\geq 3$, $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{\Theta(d)}$ holds for a large class of graphs containing, among others, all regular graphs and all graphs of bounded clique number. This settles the conjecture (up to constants) for these classes of graphs. In the general case, we prove $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{\Theta(d^{3})}$. In a second part, we prove that in any graph $G$ of minimum degree $\delta$ and girth at least 5, $\gamma^{\text{ID}}(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n$. Using the former result, we give sharp estimates for the size of the minimum identifying code of random $d$-regular graphs, which is about $\tfrac{\log d}{d}n$.

TOTAL OUTER-CONNECTED DOMINATION SUBDIVISION NUMBERS IN GRAPHS

2013 ◽  
Vol 05 (03) ◽  
pp. 1350009
Author(s):  
O. FAVARON ◽  
R. KHOEILAR ◽  
S. M. SHEIKHOLESLAMI
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Connected Dominating Set ◽  
Minimum Size ◽  
Connected Domination Number ◽  
Minimum Number ◽  
Domination Subdivision Number

A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph G[V\S] induced by V\S is connected. The total outer-connected domination numberγ toc (G) is the minimum size of such a set. The total outer-connected domination subdivision number sd γ toc (G) is the minimum number of edges that must be subdivided in order to increase the total outer-connected domination number. We prove the existence of sd γ toc (G) for every connected graph G of order at least 3 and give upper bounds on it in some classes of graphs.

Preparation of integrated circuits for TEM electromigration in situ studies

1986 ◽  
Vol 44 ◽  
pp. 882-883
Author(s):  
J. V. Maskowitz ◽  
W. E. Rhoden ◽  
D. R. Kitchen ◽  
R. E. Omlor ◽  
P. F. Lloyd
Keyword(s):  
Integrated Circuits ◽  
Crystallographic Orientation ◽  
Silicon Wafers ◽  
Minimum Size ◽  
Test Vehicle ◽  
In Situ Studies ◽  
Boron Doped ◽  
Doped Silicon ◽  
Drill Holes

The fabrication of the aluminum bridge test vehicle for use in the crystallographic studies of electromigration involves several photolithographic processes, some common, while others quite unique. It is most important to start with a clean wafer of known orientation. The wafers used are 7 mil thick boron doped silicon. The diameter of the wafer is 1.5 inches with a resistivity of 10-20 ohm-cm. The crystallographic orientation is (111).Initial attempts were made to both drill and laser holes in the silicon wafers then back fill with photoresist or mounting wax. A diamond tipped dentist burr was used to successfully drill holes in the wafer. This proved unacceptable in that the perimeter of the hole was cracked and chipped. Additionally, the minimum size hole realizable was > 300 μm. The drilled holes could not be arrayed on the wafer to any extent because the wafer would not stand up to the stress of multiple drilling.

Transplantation of coral fragment, Acropora formosa (Scleractinia)

2014 ◽  
pp. 1
Author(s):  
Hanny Tioho ◽  
Maykel A.J Karauwan
Keyword(s):  
Fragment Size ◽  
Observation Time ◽  
Minimum Size ◽  
Average Growth Rate ◽  
Average Growth ◽  
Significant Difference ◽  
Acropora Formosa ◽  
Survival And Growth ◽  
One Way Anova ◽  
Better Than

The minimum size of coral transplants, Acropora formosa, was assessed to support their survival and growth. For this, 150 coral fragments of different sizes (5, 10, 15 cm) were transplanted close to the donor colony. Their survivorship and growth were observed for 12 months. At the end of the observation time, 90% of 15 cm-transplanted coral fragments survived, while the others (10cm and 5 cm) did 86% and 82% respectively. The average growth rate of 5 cm-coral fragments was 0.860 cm/month, while 10 and 15 cm-fragments were 0.984 cm/month and 1.108 cm/month respectively. One-way ANOVA showed that there was significant difference (p<0.05) among the three (5, 10, 15 cm) transplant initial sizes in which the longest fragment size tended to survive longer than the smaller one.  However, the smaller transplants grew better than the bigger one, 10.318 cm/year (206%) for 5 cm-transplant, 11.803 cm/year (118%) for 10 cm-transplant, and 13.299 cm/year (89%) for 15 cm-transplant, respectively. Ukuran minimal fragmen karang Acropora formosa yang ditransplantasi diduga untuk mendukung ketahanan hidup dan pertumbuhannya. Untuk itu, 150 fragmen karang ditransplantasi ke lokasi yang berdekatan dengan koloni induknya.  Ketahanan hidup dan pertumbuhan semua fragmen karang yang ditransplantasi diamati selama 12 bulan.  Pada akhir pengamatan, 90% dari fragmen karang berukuran 15 cm yang ditransplantasi dapat bertahan hidup, sedangkan yang lainnya (ukuran 10 cm dan 5 cm) masing-masing sebesar 86% dan 82%.  Rata-rata laju pertumbuhan fragmen karang dengan ukuran awal 5 cm adalah 0,860 cm/bulan, sedangkan ukuran fragmen 10 dan 15 cm masing-masing adalah 0,984 cm/bulan and 1,108 cm/bulan. ANOVA satu arah menunjukkan adanya perbedaan yang nyata (p<0.05) antara ketiga ukuran fragmen yang berbeda, di mana ukuran fragmen karang yang lebih panjang cenderung mempunyai ketahanan hidup yang lebih baik. Namun demikian, ukuran transplant yang lebih kecil memiliki pertumbuhan lebih baik dibandingkan dengan ukuran yang lebih besar, yakni10,318 cm/tahun (206%) untuk transplant berukuran 5 cm, 11,803 cm/tahun (118%) untuk 10 cm, dan 13,299 cm/tahun (89%) untuk ukuran 15 cm.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

Formulation, Optimization,and Ex-Vivo Evaluation of Novel Lipid Carriers for Enhanced Transdermal Delivery of Hydroquinone

2020 ◽  
Vol 12 ◽  
Author(s):  
Shivani Verma ◽  
Sukhjinder Kaur ◽  
Lalit Kumar
Keyword(s):  
Ex Vivo ◽  
Skin Permeation ◽  
Topical Delivery ◽  
Gelling Agent ◽  
Minimum Size ◽  
Prolonged Release ◽  
Freeze Dried ◽  
Ft Ir ◽  
Ir Analysis ◽  
Skin Retention

Background: HQ is used for hyper-pigmentation treatment using conventional creams and gels. These formulations show various disadvantages like poor skin permeation, allergic reactions, and repeated use decreasing patient compliance. Objectives: The present work involved formulation, statistical optimization, and characterization of nanostructured lipid carriers (NLCs) for efficient topical delivery of hydroquinone (HQ) for hyperpigmentation treatment. Methods: The NLCs were optimized exploring Box–Behnken design (BBD) using three independent variables and two dependent variables. Formulation having the minimum size and maximum drug entrapment was considered as optimized formulation. Optimized formulation was evaluated for drug release followed by its freeze-drying. The freeze-dried formulation was subjected to differential scanning calorimetry (DSC) analysis, X-raydiffraction (XRD) analysis, and Fourier transform-infrared spectroscopy (FT-IR) analysis. Furthermore, NLCs based gel was prepared by using Carbopol 934 as a gelling agent. NLCs based gel was evaluated for skin permeation, skin retention, and skin distribution (through confocal microscopic analysis) using pig ear skin. Results: Optimized NLCs showed smaller particle size [(271.9 ± 9) nm], high drug entrapment [(66.4 ± 1.2) %], tolerable polydispersity index (PDI) (0.221 ± 0.012), and zeta potential [(-25.9± 1.2) mV]. The FT-IR analysis revealed excellent compatibility between HQ and other excipients. The Carbopol 934 gel containing NLCs showed high transdermal flux [(163 ± 16.2) μg/cm2/h], permeability coefficient (0.0326 ± 0.0016), and skin permeation enhancement ratio (3.7 ± 0.4) compared to marketed cream of HQ. The results of confocal microscopic (CLSM) analysis revealed the accumulation of optimized NLCs in the lower epidermal layers of skin. Conclusion: NLCs based gel was considered effective in the topical delivery of HQ to treat hyper-pigmentation due high skin permeation, skin retention, and prolonged release of HQ.

The Oxford Handbook of IPOs

2018 ◽  
Keyword(s):  
United States ◽  
Initial Public Offerings ◽  
Financial Regulation ◽  
Large Body ◽  
Public Information ◽  
The United States ◽  
Minimum Size ◽  
Stock Exchanges ◽  
Public Offerings ◽  
Firm’S Life Cycle

Firms generally begin as privately owned entities. When they grow large enough, the decision to go public and its consequences are among the most crucial times in a firm’s life cycle. The first time a firm is a reporting issuer gives rise to tremendous responsibilities about disclosing public information and accountability to a wide array of retail shareholders and institutional investors. Initial public offerings (IPOs) offer tremendous opportunities to raise capital. The economic and legal landscape for IPOs has been rapidly evolving across countries. There have been fewer IPOs in the United States in the aftermath of the 2007–2009 financial crisis and associated regulatory reforms that began in 2002. In 1980–2000, an average of 310 firms went public every year, while in 2001–2014 an average of 110 firms went public every year. At the same time, there are so many firms that seek an IPO in China that there has been a massive waiting list of hundreds of firms in recent years. Some countries are promoting small junior stock exchanges to go public early, and even crowdfunding to avoid any prospectus disclosure. Financial regulation of analysts and investment banks has been evolving in ways that drastically impact the economics of going public—in some countries, such as the United States, drastically increasing the minimum size of a company before it can expect to go public. This Handbook not only systematically and comprehensively consolidates a large body of literature on IPOs, but provides a foundation for future debates and inquiry.

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