scholarly journals The linear $ k $-arboricity of symmetric directed trees

2021 ◽  
Vol 7 (2) ◽  
pp. 1603-1614
Author(s):  
Xiaoling Zhou ◽  
◽  
Chao Yang ◽  
Weihua He ◽  
Keyword(s):  
Directed Path ◽  
Connected Component ◽  
Minimum Number

<abstract><p>A linear $ k $-diforest is a directed forest in which every connected component is a directed path of length at most $ k $. The linear $ k $-arboricity of a digraph $ D $ is the minimum number of arc-disjoint linear $ k $-diforests whose union covers all the arcs of $ D $. In this paper, we study the linear $ k $-arboricity for symmetric directed trees and fully determine the linear $ 2 $-arboricity for all symmetric directed trees.</p></abstract>

scholarly journals The linear $ k $-arboricity of digraphs

2021 ◽  
Vol 7 (3) ◽  
pp. 4137-4152
Author(s):  
Xiaoling Zhou ◽  
◽  
Chao Yang ◽  
Weihua He ◽  
Keyword(s):  
Directed Path ◽  
Connected Component ◽  
Minimum Number

<abstract><p>A linear $ k $-diforest is a directed forest in which every connected component is a directed path of length at most $ k $. The linear $ k $-arboricity of a digraph $ D $ is the minimum number of linear $ k $-diforests needed to partition the arcs of $ D $. In this paper, we study the linear $ k $-arboricity for digraphs, and determine the linear $ 3 $-arboricity and linear $ 2 $-arboricity for symmetric complete digraphs and symmetric complete bipartite digraphs.</p></abstract>

scholarly journals Extension of Gyárfás-Sumner Conjecture to Digraphs

10.37236/9906 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Pierre Aboulker ◽  
Pierre Charbit ◽  
Reza Naserasr
Keyword(s):  
Chromatic Number ◽  
Directed Path ◽  
Complete Multipartite Graph ◽  
Color Class ◽  
Acyclic Digraph ◽  
Minimum Number ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Dichromatic Number

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices  in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization  of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph.  Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

The Restricted Edge-Connectivity of Kronecker Product Graphs

2019 ◽  
Vol 29 (03) ◽  
pp. 1950012
Author(s):  
Tianlong Ma ◽  
Jinling Wang ◽  
Mingzu Zhang
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Kronecker Product ◽  
Sufficient Condition ◽  
Connected Component ◽  
Edge Connectivity ◽  
Product Graphs ◽  
Minimum Number ◽  
Vertex Set ◽  
Product Of Graphs

The restricted edge-connectivity of a connected graph [Formula: see text], denoted by [Formula: see text], if exists, is the minimum number of edges whose deletion disconnects the graph such that each connected component has at least two vertices. The Kronecker product of graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text], where two vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. In this paper, it is proved that [Formula: see text] for any graph [Formula: see text] and a complete graph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is minimum edge-degree of [Formula: see text], and a sufficient condition such that [Formula: see text] is [Formula: see text]-optimal is acquired.

scholarly journals Covering Small Subgraphs of (Hyper)Tournaments with Spanning Acyclic Subgraphs

10.37236/9336 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Raphael Yuster
Keyword(s):  
Lower Bound ◽  
Total Order ◽  
Exact Order ◽  
Directed Path ◽  
Minimum Number ◽  
Order Of Magnitude

While the edges of every tournament can be covered with two spanning acyclic subgraphs, this is not so if we set out to cover all acyclic $H$-subgraphs of a tournament with spanning acyclic subgraphs, even for very simple $H$ such as the $2$-edge directed path or the $2$-edge out-star. We prove new bounds for the minimum number of elements in such coverings and for some $H$ our bounds determine the exact order of magnitude. A $k$-tournament is an orientation of the complete $k$-graph, where each $k$-set is given a total order (so tournaments are $2$-tournaments). As opposed to tournaments, already covering the edges of a $3$-tournament with the minimum number of spanning acyclic subhypergraphs is a nontrivial problem. We prove a new lower bound for this problem which asymptotically matches the known lower bound of covering all ordered triples of a set.

Information capacity and bandwidth limitations in the SEM

1989 ◽  
Vol 47 ◽  
pp. 84-85
Author(s):  
D. C. Joy ◽  
R. D. Bunn
Keyword(s):  
Signal To Noise Ratio ◽  
Critical Dimension ◽  
Information Capacity ◽  
Acquisition Time ◽  
Signal To Noise ◽  
Sem Image ◽  
Probe Size ◽  
Minimum Number ◽  
Noise Ratio ◽  
Signal Channel

The information available from an SEM image is limited both by the inherent signal to noise ratio that characterizes the image and as a result of the transformations that it may undergo as it is passed through the amplifying circuits of the instrument. In applications such as Critical Dimension Metrology it is necessary to be able to quantify these limitations in order to be able to assess the likely precision of any measurement made with the microscope.The information capacity of an SEM signal, defined as the minimum number of bits needed to encode the output signal, depends on the signal to noise ratio of the image - which in turn depends on the probe size and source brightness and acquisition time per pixel - and on the efficiency of the specimen in producing the signal that is being observed. A detailed analysis of the secondary electron case shows that the information capacity C (bits/pixel) of the SEM signal channel could be written as :

Applying Generalizability Theory to Optimize Analysis of Spontaneous Teacher Talk in Elementary Classrooms

2020 ◽  
Vol 63 (6) ◽  
pp. 1947-1957
Author(s):  
Alexandra Hollo ◽  
Johanna L. Staubitz ◽  
Jason C. Chow
Keyword(s):  
Special Education Teachers ◽  
Generalizability Theory ◽  
Child Outcomes ◽  
Elementary Classrooms ◽  
Teacher Talk ◽  
Group Instruction ◽  
Data Set ◽  
School Year ◽  
Minimum Number ◽  
Representative Samples

Purpose Although sampling teachers' child-directed speech in school settings is needed to understand the influence of linguistic input on child outcomes, empirical guidance for measurement procedures needed to obtain representative samples is lacking. To optimize resources needed to transcribe, code, and analyze classroom samples, this exploratory study assessed the minimum number and duration of samples needed for a reliable analysis of conventional and researcher-developed measures of teacher talk in elementary classrooms. Method This study applied fully crossed, Person (teacher) × Session (samples obtained on 3 separate occasions) generalizability studies to analyze an extant data set of three 10-min language samples provided by 28 general and special education teachers recorded during large-group instruction across the school year. Subsequently, a series of decision studies estimated of the number and duration of sessions needed to obtain the criterion g coefficient ( g > .70). Results The most stable variables were total number of words and mazes, requiring only a single 10-min sample, two 6-min samples, or three 3-min samples to reach criterion. No measured variables related to content or complexity were adequately stable regardless of number and duration of samples. Conclusions Generalizability studies confirmed that a large proportion of variance was attributable to individuals rather than the sampling occasion when analyzing the amount and fluency of spontaneous teacher talk. In general, conventionally reported outcomes were more stable than researcher-developed codes, which suggests some categories of teacher talk are more context dependent than others and thus require more intensive data collection to measure reliably.

RGB Frequency Based Image Steganography

2020 ◽  
pp. 549-553
Author(s):  
Himanshu Kumar ◽  
Nitesh Kumar
Keyword(s):  
Image Steganography ◽  
Minimum Number

In this paper, we introduced a new RGB technique for image steganography. In this technique we introduced the idea of storing a different number of bits per channel (R, G or B) of a pixel based on the frequency of color values of pixel. The higher color frequency retains the maximum number of bits and lower color frequency stores the minimum number of bits.

On Number of Planes of Rearrangeably Nonblocking Optical Banyan Networks with Link Failures

2008 ◽  
Vol 1 (1) ◽  
pp. 43-54
Author(s):  
Basra Sultana ◽  
Mamun-ur-Rashid Khandker
Keyword(s):  
Optical Switching ◽  
Blocking Probability ◽  
Link Failure ◽  
Link Failures ◽  
Small Depth ◽  
Internal Link ◽  
Banyan Network ◽  
Minimum Number ◽  
Switch Design ◽  
Banyan Networks

Vertically stacked optical banyan (VSOB) networks are attractive for serving as optical switching systems due to the desirable properties (such as the small depth and self-routing capability) of banyan network structures. Although banyan-type networks result in severe blocking and crosstalk, both these problems can be minimized by using sufficient number of banyan planes in the VSOB network structure. The number of banyan planes is minimum for rearrangeably nonblocking and maximum for strictly nonblocking structure. Both results are available for VSOB networks when there exist no internal link-failures. Since the issue of link-failure is unavoidable, we intend to find the minimum number of planes required to make a VSOB network nonblocking when some links are broken or failed in the structure. This paper presents the approximate number of planes required to make a VSOB networks rearrangeably nonblocking allowing link-failures. We also show an interesting behavior of the  blocking  probability of a faulty VSOB networks that the blocking probability may not  always  increase monotonously with  the  increase  of  link-failures; blocking probability  decreases  for  certain range of  link-failures, and then increases again. We believe that such fluctuating behavior of blocking probability with the increase of link failure probability deserves special attention in switch design.  Keywords: Banyan networks; Blocking probability; Switching networks; Vertical stacking; Link-failures. © 2009 JSR Publications. ISSN: 2070-0237(Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v1i1.1070

Mathematics of a Sudo-Kurve

2018 ◽  
Vol 5 (10) ◽  
pp. 5-28
Author(s):  
Tanya Khovanova ◽  
Wayne Zhao
Keyword(s):  
Minimum Number ◽  
Distinct Solution ◽  
Equivalent Variant

Abstract We investigate a type of a Sudoku variant called Sudo-Kurve, which allows bent rows and columns, and develop a new, yet equivalent, variant we call a Sudo-Cube. We examine the total number of distinct solution grids for this type with or without symmetry. We study other mathematical aspects of this puzzle along with the minimum number of clues needed and the number of ways to place individual symbols.

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