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 $n\geq 3$, $\rho(K_3^n)\in \{ \left\lceil {\log_3 (2n+1)} \right\rceil$ $+1, \left\lceil {\log_3 (2n+1)} \right\rceil$ $+2\}$.

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

M-Series with Adaptive Patterning for High-Yield Fan-Out SIP

2016 ◽  
Vol 2016 (DPC) ◽  
pp. 000751-000773
Author(s):  
Craig Bishop ◽  
Suresh Jayaraman ◽  
Boyd Rogers ◽  
Chris Scanlan ◽  
Tim Olson
Keyword(s):  
New Technology ◽  
Minimum Size ◽  
High Yield ◽  
Wafer Level ◽  
Design Rules ◽  
Optical Scanner ◽  
Wafer Level Packaging ◽  
And Performance ◽  
The Cost ◽  
Semiconductor Chips

Fan-Out Wafer Level Packaging (FOWLP) holds immediate promise for packaging semiconductor chips with higher interconnect density than the incumbent Wafer Level Chip Scale Packaging (WLCSP). FOWLP enables size and performance capabilities similar to WLCSP, while extending capabilities to include multi-device system-in-packages. FOWLP can support applications that integrate multiple heterogeneously processed die at lower cost than 2.5D silicon interposer technologies. Current industry challenges with die position yield after die placement and molding result in low-density design rules and the high-cost of accurate die placement. Efficiently handling die shift is essential for making FOWLP cost-competitive with other technologies such as FCCSP and QFN. This presentation will provide an overview of Adaptive Patterning, a new technology for overcoming variability of die positions after placement and molding. In this process, an optical scanner is used to measure the true XY position and rotation of each die after panelization. The die measurements are then fed into a proprietary software engine that generates a unique pattern for each package. The resulting patterns are dispatched to a lithography system, which dynamically implements the unique patterns for all packages within a panel. For system-in-packages, this process offers a unique advantage over a fixed pattern: each die shift can be handled independently. With a fixed pattern, the design tolerances need to be large enough for all die to shift in opposing directions, otherwise yield loss in incurred. With Adaptive Patterning, vias and RDL features remain at minimum size and are matched to the measured die shift. The die-to-die interconnects are dynamically generated and account for the unique position of each die. Thus, Adaptive Patterning retains the same high-density design rules regardless of how many die are in a package. Adaptive Patterning provides the capability to use high-throughput die placement to drive down cost, while enabling higher-density system-in-package interconnect. With this technology the industry can finally realize the cost, flexibility, and form factor benefits of FOWLP.

scholarly journals Cost Effective Domination in the Join, Corona and Composition of Graphs

2019 ◽  
Vol 12 (3) ◽  
pp. 978-998
Author(s):  
Ferdinand P. Jamil ◽  
Hearty Nuenay Maglanque
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Cost Effective ◽  
Minimal Cost ◽  
Dominating Sets ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
The Cost ◽  
Domination Numbers

Let $G$ be a connected graph. A cost effective dominating set in a graph $G$ is any set $S$ of vertices in $G$ satisfying the condition that each vertex in $S$ is adjacent to at least as many vertices outside $S$ as inside $S$ and every vertex outside $S$ is adjacent to at least one vertex in $S$. The minimum cardinality of a cost effective dominating set is the cost effective domination number of $G$. The maximum cardinality of a cost effective dominating set is the upper cost effective domination number of $G$. A cost effective dominating set is said to be minimal if it does not contain a proper subset which is itself a cost effective dominating in $G$. The maximum cardinality of a minimal cost effective dominating set in a graph $G$ is the minimal cost effective domination number of $G$.In this paper, we characterized the cost effective dominating sets in the join, corona and composition of graphs. As direct consequences, we the bounds or the exact cost effective domination numbers, minimal cost effective domination numbers and upper cost effective domination numbers of these graphs were obtained.

scholarly journals Minimal and Upper Cost Effective Domination in Graphs

2021 ◽  
Vol 14 (2) ◽  
pp. 537-550
Author(s):  
Hearty Nuenay Maglanque ◽  
Ferdinand P. Jamil
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Cost Effective ◽  
Minimal Cost ◽  
Type Result ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
The Cost ◽  
Domination In Graphs

Given a connected graph $G$, we say that $S\subseteq V(G)$ is a cost effective dominating set in $G$ if, each vertex in $S$ is adjacent to at least as many vertices outside $S$ as inside $S$ and that every vertex outside $S$ is adjacent to at least one vertex in $S$. The minimum cardinality of a cost effective dominating set is the cost effective domination number of $G$. The maximum cardinality of a cost effective dominating set is the upper cost effective domination number of $G$, and is denoted by $\gamma_{ce}^+(G).$ A cost effective dominating set is said to be minimal if it does not contain a proper subset which is itself a cost effective dominating in $G$. The maximum cardinality of a minimal cost effective dominating set in a graph $G$ is the minimal cost effective domination number of $G$, and is denoted by $\gamma_{mce}(G)$. In this paper we provide bounds on upper cost effective domination number and minimal cost effective domination number of a connected graph G and characterized those graphs whose upper and minimal cost effective domination numbers are either $1, 2$ or $n-1.$ We also establish a Nordhaus-Gaddum type result for the introduced parameters and solve some realization problems.

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.

scholarly journals Infinite Graphs with Finite 2-Distinguishing Cost

10.37236/4263 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Debra Boutin ◽  
Wilfried Imrich
Keyword(s):  
Automorphism Group ◽  
Linear Growth ◽  
Infinite Graphs ◽  
Minimum Size ◽  
Locally Finite ◽  
Determining Set ◽  
The Cost

A graph $G$ is said to be 2-distinguishable if there is a labeling of the vertices with two labels such that only the trivial automorphism preserves the labels. Call the minimum size of a label class in such a labeling of $G$ the cost of 2-distinguishing $G$.We show that the connected, locally finite, infinite graphs with finite 2-distinguishing cost are exactly the graphs with countable automorphism group. Further we show that in such graphs the cost is less than three times the size of a smallest determining set. We also another, sharper bound on the 2-distinguishing cost, in particular for graphs of linear growth.

The IBM-PC in electron microscopy

1996 ◽  
Vol 54 ◽  
pp. 612-613
Author(s):  
James F. Mancuso
Keyword(s):  
Head Start ◽  
Image Acquisition ◽  
Cost Effective ◽  
Financial Applications ◽  
Number Of Factors ◽  
Instrument Control ◽  
The Cost ◽  
Training And Support ◽  
Control Image ◽  
Ibm Pc

IBM PC compatible computers are widely used in microscopy for applications ranging from control to image acquisition and analysis. The choice of IBM-PC based systems over competing computer platforms can be based on technical merit alone or on a number of factors relating to economics, availability of peripherals, management dictum, or simple personal preference.IBM-PC got a strong “head start” by first dominating clerical, document processing and financial applications. The use of these computers spilled into the laboratory where the DOS based IBM-PC replaced mini-computers. Compared to minicomputer, the PC provided a more for cost-effective platform for applications in numerical analysis, engineering and design, instrument control, image acquisition and image processing. In addition, the sitewide use of a common PC platform could reduce the cost of training and support services relative to cases where many different computer platforms were used. This could be especially true for the microscopists who must use computers in both the laboratory and the office.

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