Excluding Minors in Nonplanar Graphs of Girth at Least Five

2000 ◽  
Vol 9 (6) ◽  
pp. 573-585 ◽  
Author(s):  
ROBIN THOMAS ◽  
JAN McDONALD THOMSON
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
The Other ◽  
Petersen Graph ◽  
Regular Graphs ◽  
Nonplanar Graph ◽  
A Minor

A graph G is quasi 4-connected if it is simple, 3-connected, has at least five vertices, and for every partition (A, B, C) of V(G) either [mid ]C[mid ] [ges ] 4, or G has an edge with one end in A and the other end in B, or one of A,B has at most one vertex. We show that any quasi 4-connected nonplanar graph with minimum degree at least three and no cycle of length less than five has a minor isomorphic to P−10, the Petersen graph with one edge deleted. We deduce the following weakening of Tutte's Four Flow Conjecture: every 2-edge-connected graph with no minor isomorphic to P−10 has a nowhere-zero 4-flow. This extends a result of Kilakos and Shepherd who proved the same for 3-regular graphs.

scholarly journals Colouring the Petals of a Graph

10.37236/1699 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
David Cariolaro ◽  
Gianfranco Cariolaro
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Petersen Graph ◽  
Empty Graph ◽  
Single Exception ◽  
Class 1

A petal graph is a connected graph $G$ with maximum degree three, minimum degree two, and such that the set of vertices of degree three induces a $2$–regular graph and the set of vertices of degree two induces an empty graph. We prove here that, with the single exception of the graph obtained from the Petersen graph by deleting one vertex, all petal graphs are Class $1$. This settles a particular case of a conjecture of Hilton and Zhao.

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 $\tfrac{\log d}{d}n$.

scholarly journals On $k$-Walk-Regular Graphs

10.37236/136 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
C. Dalfó ◽  
M. A. Fiol ◽  
E. Garriga
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
The Other ◽  
Regular Graphs ◽  
Local Spectrum ◽  
Number Of Cycles

Considering a connected graph $G$ with diameter $D$, we say that it is $k$-walk-regular, for a given integer $k$ $(0\leq k \leq D)$, if the number of walks of length $\ell$ between any pair of vertices only depends on the distance between them, provided that this distance does not exceed $k$. Thus, for $k=0$, this definition coincides with that of walk-regular graph, where the number of cycles of length $\ell$ rooted at a given vertex is a constant through all the graph. In the other extreme, for $k=D$, we get one of the possible definitions for a graph to be distance-regular. In this paper we show some algebraic characterizations of $k$-walk-regularity, which are based on the so-called local spectrum and predistance polynomials of $G$.

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

A modified Runge-Kutta method

SIMULATION ◽  
1968 ◽  
Vol 10 (5) ◽  
pp. 221-223 ◽  
Author(s):  
A.S. Chai
Keyword(s):  
Error Estimate ◽  
Linear Combination ◽  
Experimental Results ◽  
The Other ◽  
New Method ◽  
Kutta Method ◽  
Starting Values ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
A Minor

It is possible to replace k2 in a 4th-order Runge-Kutta for mula (also Nth-order 3 ≤ N ≤ 5) by a linear combination of k1 and the ki's in the last step, using the same procedure for computing the other ki's and y as in the standard R-K method. The advantages of the new method are: It re quires one less derivative evaluation, provides an error estimate at each step, gives more accurate results, and needs a minor change to switch to the RK to obtain the starting values. Experimental results are shown in verification of the for mula.

scholarly journals II.—General Theorems on Determinants

1880 ◽  
Vol 29 (1) ◽  
pp. 47-54
Author(s):  
Thomas Muir
Keyword(s):  
The Other ◽  
Elementary Theorem ◽  
A Minor

The rows of a determinant of the nth order having been separated into two sets, one containing the first p rows and the other the rest, if each minor of the pth degree formed from the first set be multiplied by a minor, called its complementary, formed from the second set, and the result have its sign chosen in accordance with a certain law, it is well known as an elementary theorem that the aggregate of the products thus obtained is equal to the original determinant.

A Medieval Open Work

POETICA ◽  
2021 ◽  
Vol 52 (3-4) ◽  
pp. 228-265
Author(s):  
Rafael Simian
Keyword(s):  
The Other ◽  
Specific Function ◽  
Umberto Eco ◽  
New Approach ◽  
Avant Garde ◽  
Other Hand ◽  
Open Work ◽  
New Perspective ◽  
A Minor

Abstract Guigo II is commonly known and praised among specialists of Western mysticism for his Scala claustralium, a work that presents a spiritual program for cloistered monks. His Meditations, on the other hand, have usually been relegated to the margin of attention. The First Meditation, in particular, is generally regarded as a minor piece. The paper argues, however, that a new approach can make better sense of the First Meditation, while also enabling us to recognize its specific function and value. Seen from this new perspective, Guigo’s purpose with the text is to train and exercise his readers’ minds according to the spiritual program laid out in the Scala. The paper shows that the First Meditation realizes that goal, surprisingly, by having the same essential features that Umberto Eco found in the ‘open works’ of the Western avant-garde.

Scalable mining of maximal quasi-cliques

2020 ◽  
Vol 14 (4) ◽  
pp. 573-585
Author(s):  
Guimu Guo ◽  
Da Yan ◽  
M. Tamer Özsu ◽  
Zhe Jiang ◽  
Jalal Khalil
Keyword(s):  
Graph Mining ◽  
State Of The Art ◽  
Minimum Degree ◽  
The Other ◽  
Dense Subgraph ◽  
Load Imbalance ◽  
Subgraph Mining ◽  
Execution Engine ◽  
Clique Algorithm ◽  
Dense Subgraph Mining

Given a user-specified minimum degree threshold γ , a γ -quasiclique is a subgraph g = (V g , E g ) where each vertex ν ∈ V g connects to at least γ fraction of the other vertices (i.e., ⌈ γ · (| V g |- 1)⌉ vertices) in g. Quasi-clique is one of the most natural definitions for dense structures useful in finding communities in social networks and discovering significant biomolecule structures and pathways. However, mining maximal quasi-cliques is notoriously expensive. In this paper, we design parallel algorithms for mining maximal quasi-cliques on G-thinker, a distributed graph mining framework that decomposes mining into compute-intensive tasks to fully utilize CPU cores. We found that directly using G-thinker results in the straggler problem due to (i) the drastic load imbalance among different tasks and (ii) the difficulty of predicting the task running time. We address these challenges by redesigning G-thinker's execution engine to prioritize long-running tasks for execution, and by utilizing a novel timeout strategy to effectively decompose long-running tasks to improve load balancing. While this system redesign applies to many other expensive dense subgraph mining problems, this paper verifies the idea by adapting the state-of-the-art quasi-clique algorithm, Quick, to our redesigned G-thinker. Extensive experiments verify that our new solution scales well with the number of CPU cores, achieving 201× runtime speedup when mining a graph with 3.77M vertices and 16.5M edges in a 16-node cluster.

COMPUTING THE RUPTURE DEGREE IN COMPOSITE GRAPHS

2010 ◽  
Vol 21 (03) ◽  
pp. 311-319 ◽  
Author(s):  
AYSUN AYTAC ◽  
ZEYNEP NIHAN ODABAS
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
The Other ◽  
Trade Off ◽  
Number Of Components ◽  
Np Complete ◽  
Work Done ◽  
Better Than

The rupture degree of an incomplete connected graph G is defined by [Formula: see text] where w(G - S) is the number of components of G - S and m(G - S) is the order of a largest component of G - S. For the complete graph Kn, rupture degree is defined as 1 - n. This parameter can be used to measure the vulnerability of a graph. Rupture degree can reflect the vulnerability of graphs better than or independent of the other parameters. To some extent, it represents a trade-off between the amount of work done to damage the network and how badly the network is damaged. Computing the rupture degree of a graph is NP-complete. In this paper, we give formulas for the rupture degree of composition of some special graphs and we consider the relationships between the rupture degree and other vulnerability parameters.

Paranormal Belief, Experience, and the Keirsey Temperament Sorter

2000 ◽  
Vol 86 (3_part_2) ◽  
pp. 1104-1106
Author(s):  
Jezz Fox ◽  
Carl Williams
Keyword(s):  
College Students ◽  
Multiple Regression ◽  
The Other ◽  
Minor Role ◽  
Regression Analyses ◽  
Keirsey Temperament Sorter ◽  
Paranormal Belief ◽  
Anomalous Experience ◽  
A Minor ◽  
Paranormal Experience

121 college students completed the Anomalous Experience Inventory and the Keirsey Temperament Sorter. Multiple regression analyses provided significant models predicting both Paranormal Experience and Belief; the main predictors were the other subscales of the Anomalous Experience Inventory with the Keirsey variables playing only a minor role.

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