Graphs with 6-Ways

1973 ◽  
Vol 25 (4) ◽  
pp. 687-692 ◽  
Author(s):  
John L. Leonard
Keyword(s):  
Finite Graph ◽  
Minimum Number ◽  
Multiple Edges

In a finite graph with no loops nor multiple edges, two points a and b are said to be connected by an r-way, or more explicitly, by a line r-way a — b if there are r paths, no two of which have lines in common (although they may share common points), which join a to b. In this note we demonstrate that any graph with n points and 3n — 2 or more lines must contain a pair of points joined by a 6-way, and that 3n — 2 is the minimum number of lines which guarantees the presence of a 6-way in a graph of n points.In the language of [3], this minimum number of lines needed to guarantee a 6-way is denoted U(n). For the background of this problem, the reader is referred to [3].

scholarly journals A VISIBILITY-BASED PURSUIT-EVASION PROBLEM

1999 ◽  
Vol 09 (04n05) ◽  
pp. 471-493 ◽  
Author(s):  
LEONIDAS J. GUIBAS ◽  
JEAN-CLAUDE LATOMBE ◽  
STEVEN M. LAVALLE ◽  
DAVID LIN ◽  
RAJEEV MOTWANI
Keyword(s):  
Finite Graph ◽  
Initial Position ◽  
Moving Target ◽  
Multiply Connected ◽  
Pursuit Evasion ◽  
Minimum Number ◽  
Evasion Problem ◽  
Free Spaces ◽  
Simply Connected ◽  
Motion Strategy

This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our study of this problem is motivated in part by robotics applications, such as surveillance with a mobile robot equipped with a camera that must find a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy for a single pursuer. For simply-connected free spaces, it is shown that the minimum number of pursuers required is Θ( lg  n). For multiply-connected free spaces, the bound is [Formula: see text] pursuers for a polygon that has n edges and h holes. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a finite graph that is constructed on the basis of critical information changes. It has been implemented and computed examples are shown.

scholarly journals A note on the conditional chromatic polynomial

1994 ◽  
Vol 36 (3) ◽  
pp. 265-267 ◽  
Author(s):  
V. Voloshin
Keyword(s):  
Finite Graph ◽  
Chromatic Polynomial ◽  
Multiple Edges

In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in λ colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed λ [1, 4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another.

Triangles in an Ordinary Graph

1963 ◽  
Vol 15 ◽  
pp. 33-41 ◽  
Author(s):  
E. A. Nordhaus ◽  
B. M. Stewart
Keyword(s):  
Linear Graph ◽  
Minimum Number ◽  
Multiple Edges ◽  
Finite Linear

An ordinary graph is a finite linear graph which contains no loops or multiple edges, and in which all edges are undirected. In such a graph G, let N, L, and T denote respectively the number of nodes, edges, and triangles. One problem, suggested by P. Erdös (1), is to determine the minimum number of triangles when the number of edges is specified, subject to suitable restrictions.

scholarly journals Finding a Princess in a Palace: a Pursuit-Evasion Problem

10.37236/2296 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
John R Britnell ◽  
Mark Wildon
Keyword(s):  
Winning Strategy ◽  
Finite Graph ◽  
Pursuit Evasion ◽  
Single Room ◽  
Cops And Robbers ◽  
Minimum Number ◽  
Evasion Problem

This paper solves a pursuit-evasion problem in which a prince must find a princess who is constrained to move on each day from one vertex of a finite graph to another. Unlike the related and much studied `Cops and Robbers Game', the prince has no knowledge of the position of the princess; he may, however, visit any single room he wishes on each day. We characterize the graphs for which the prince has a winning strategy, and determine, for each such graph, the minimum number of days the prince requires to guarantee to find the princess.

scholarly journals Power Dominator Chromatic Number for some Special Graphs

2019 ◽  
Vol 8 (12) ◽  
pp. 3957-3960
Keyword(s):  
Chromatic Number ◽  
Star Graph ◽  
Proper Coloring ◽  
Induction Method ◽  
Color Class ◽  
Wheel Graph ◽  
Minimum Number ◽  
Fan Graph ◽  
Multiple Edges ◽  
Dominator Coloring

Let G = (V, E) be a finite, connected, undirected with no loops, multiple edges graph. Then the power dominator coloring of G is a proper coloring of G, such that each vertex of G power dominates every vertex of some color class. The minimum number of color classes in a power dominator coloring of the graph, is the power dominator chromatic number . Here we study the power dominator chromatic number for some special graphs such as Bull Graph, Star Graph, Wheel Graph, Helm graph with the help of induction method and Fan Graph. Suitable examples are provided to exemplify the results.

Extremal Point and Edge Sets in n-Graphs

1969 ◽  
Vol 21 ◽  
pp. 1069-1075
Author(s):  
N. Sauer
Keyword(s):  
Finite Graph ◽  
Extremal Point ◽  
Minimal Cover ◽  
Covering Problems ◽  
Graph Covering ◽  
Minimum Number ◽  
Image Position ◽  
Set Of Points

A set of points (edges) of a graph is independent if no two distinct members of the set are adjacent. Gallai (1) observed that, if A0 (B0) is the minimum number of points (edges) of a finite graph covering all the edges (points) and A1 (B1) is the maximum number of independent points (edges), then:holds, where m is the number of points of the graph.The concepts of independence and covering are generalized in various ways for n-graphs. In this paper we establish certain connections between the corresponding extreme numbers analogous to the above result of Gallai.Ray-Chaudhuri considered (2) independence and covering problems in n-graphs and determined algorithms for finding the minimal cover and some associated numbers.

A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function

2021 ◽  
Vol 71 (3) ◽  
pp. 757-772
Author(s):  
Irina Gelbukh
Keyword(s):  
Fundamental Group ◽  
Height Function ◽  
Finite Graph ◽  
Critical Values ◽  
Reeb Graph ◽  
Special Cases ◽  
Cycle Rank ◽  
Multiple Edges

Abstract We prove that a finite graph (allowing loops and multiple edges) is homeomorphic (isomorphic up to vertices of degree two) to the Reeb graph of a Morse–Bott function on a smooth closed n-manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with any number k ≥ 3 of critical values, and in a few special cases with k < 3. In the case of surfaces, the function can be chosen, except for a few special cases, as the height function associated with an immersion ℝ3.

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.

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