scholarly journals The maximum size of an independent set in a nonplanar graph

1975 ◽  
Vol 81 (3) ◽  
pp. 554-556 ◽  
Author(s):  
Michael O. Albertson ◽  
Joan P. Hutchinson
Keyword(s):  
Independent Set ◽  
Maximum Size ◽  
Nonplanar Graph

scholarly journals ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

2021 ◽  
pp. 1-40
Author(s):  
NICK GILL ◽  
BIANCA LODÀ ◽  
PABLO SPIGA
Keyword(s):  
Permutation Group ◽  
Model Theory ◽  
Independent Set ◽  
Maximum Size ◽  
Proper Subset ◽  
Permutation Groups ◽  
Finite Permutation Group ◽  
Pointwise Stabilizer ◽  
Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

MAXIMUM INDEPENDENT SET OF A PERMUTATION GRAPH IN K TRACKS

1993 ◽  
Vol 03 (03) ◽  
pp. 291-304 ◽  
Author(s):  
D.T. LEE ◽  
MAJID SARRAFZADEH
Keyword(s):  
Independent Set ◽  
Maximum Size ◽  
Interval Graph ◽  
Vertical Line ◽  
Maximum Independent Set ◽  
Permutation Graph ◽  
End Points

A maximum weighted independent set of a permutation graph is a maximum subset of noncrossing chords in a matching diagram (i.e., a set Φ of chords with end-points on two horizontal lines). The problem of finding, among all noncrossing subsets of Φ with density at most k, one with maximum size is considered, where the density of a subset is the maximum number of chords crossing a vertical line and k is a given parameter. A Θ(n log n) time and Θ(n) space algorithm, for solving the problem with n chords, is proposed. As an application, we solve the problem of finding, among all proper subsets with density at most k of an interval graph, one with maximum number of intervals.

scholarly journals On independent position sets in graphs

2021 ◽  
Vol 40 (2) ◽  
pp. 385-398
Author(s):  
Elias John Thomas ◽  
Ullas Chandran S. V.
Keyword(s):  
Independent Set ◽  
Maximum Size ◽  
General Relationship ◽  
Lexicographic Product ◽  
Graph Invariants ◽  
General Properties ◽  
Product Graphs ◽  
Related Graph ◽  
Corona Product ◽  
Lexicographic Product Graphs

An independent set S of vertices in a graph G is an independent position set if no three vertices of S lie on a common geodesic. An independent position set of maximum size is an ip-set of G. The cardinality of an ip-set is the independent position number, denoted by ip(G). In this paper, we introduce and study the independent position number of a graph. Certain general properties of these concepts are discussed. Graphs of order n having the independent position number 1 or n − 1 are characterized. Bounds for the independent position number of Cartesian and Lexicographic product graphs are determined and the exact value for Corona product graphs are obtained. Finally, some realization results are proved to show that there is no general relationship between independent position sets and other related graph invariants.

Turán-Ramsey Theorems and Kp-Independence Numbers

1994 ◽  
Vol 3 (3) ◽  
pp. 297-325 ◽  
Author(s):  
P. Erdős ◽  
A. Hajnal ◽  
M. Simonovits ◽  
V. T. Sós ◽  
E. Szemerédi
Keyword(s):  
Simple Structure ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Size ◽  
Maximum Order ◽  
Induced Subgraph ◽  
Asymptotic Value ◽  
Image Position ◽  
Independence Numbers ◽  
Edge Colouring

Let the Kp-independence number αp (G) of a graph G be the maximum order of an induced subgraph in G that contains no Kp. (So K2-independence number is just the maximum size of an independent set.) For given integers r, p, m > 0 and graphs L1,…,Lr, we define the corresponding Turán-Ramsey function RTp(n, L1,…,Lr, m) to be the maximum number of edges in a graph Gn of order n such that αp(Gn) ≤ m and there is an edge-colouring of G with r colours such that the jth colour class contains no copy of Lj, for j = 1,…, r. In this continuation of [11] and [12], we will investigate the problem where, instead of α(Gn) = o(n), we assume (for some fixed p > 2) the stronger condition that αp(Gn) = o(n). The first part of the paper contains multicoloured Turán-Ramsey theorems for graphs Gn of order n with small Kp-independence number αp(Gn). Some structure theorems are given for the case αp(Gn) = o(n), showing that there are graphs with fairly simple structure that are within o(n2) of the extremal size; the structure is described in terms of the edge densities between certain sets of vertices.The second part of the paper is devoted to the case r = 1, i.e., to the problem of determining the asymptotic value offor p < q. Several results are proved, and some other problems and conjectures are stated.

ON THE IDEMPOTENT GRAPH OF A RING

2013 ◽  
Vol 12 (06) ◽  
pp. 1350003 ◽  
Author(s):  
S. AKBARI ◽  
M. HABIBI ◽  
A. MAJIDINYA ◽  
R. MANAVIYAT
Keyword(s):  
Finite Field ◽  
Division Ring ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Size ◽  
Artinian Ring ◽  
Clique Number ◽  
Infinite Field ◽  
Maximal Ideals ◽  
Reduced Ring

The idempotent graph of a ring R, denoted by I(R), is a graph whose vertices are all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = yx = 0. In this paper we show if D is a division ring, then the clique number of I(Mn(D))(n ≥ 2) is n and for any commutative Artinian ring R the clique number and the chromatic number of I(R) are equal to the number of maximal ideals of R. We prove that for every left Noetherian ring R, the clique number of I(R) is finite. For every finite field F, we also determine an independent set of I(Mn(F)) with maximum size. If F is an infinite field, then we prove that the domination number of I(Mn(F)) is infinite. We show that the idempotent graph of every reduced ring is connected and if n ≥ 3 and D is a division ring, then I(Mn(D)) is connected and moreover diam(I(Mn(D))) ≤ 5.

scholarly journals On Some Conjectures Concerning Critical Independent Sets of a Graph

10.37236/5580 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Taylor Short
Keyword(s):  
Lower Bound ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Size ◽  
Independent Sets ◽  
Simple Graph ◽  
Vertex Set ◽  
The Difference

Let $G$ be a simple graph with vertex set $V(G)$. A set $S\subseteq V(G)$ is independent if no two vertices from $S$ are adjacent. For $X\subseteq V(G)$, the difference of $X$ is $d(X) = |X|-|N(X)|$ and an independent set $A$ is critical if $d(A) = \max \{d(X): X\subseteq V(G) \text{ is an independent set}\}$ (possibly $A=\emptyset$). Let $\text{nucleus}(G)$ and $\text{diadem}(G)$ be the intersection and union, respectively, of all maximum size critical independent sets in $G$. In this paper, we will give two new characterizations of Konig-Egervary graphs involving $\text{nucleus}(G)$ and $\text{diadem}(G)$. We also prove a related lower bound for the independence number of a graph. This work answers several conjectures posed by Jarden, Levit, and Mandrescu.

Continuous Quantitative Monitoring of Mural, Platelet-Dependent, Thrombus Kinetics in the Crushed Rat Femoral Vein

1991 ◽  
Vol 65 (04) ◽  
pp. 425-431 ◽  
Author(s):  
F Stockmans ◽  
H Deckmyn ◽  
J Gruwez ◽  
J Vermylen ◽  
R Acland
Keyword(s):  
Thrombus Formation ◽  
Femoral Vein ◽  
Maximum Size ◽  
Digital Analysis ◽  
Video Recordings ◽  
Quantitative Monitoring ◽  
Set Up ◽  
Kinetics Of ◽  
Quantitative Nature

SummaryA new in vivo method to study the size and dynamics of a growing mural thrombus was set up in the rat femoral vein. The method uses a standardized crush injury to induce a thrombus, and a newly developed transilluminator combined with digital analysis of video recordings. Thrombi in this model formed rapidly, reaching a maximum size 391 ± 35 sec following injury, after which they degraded with a half-life of 197 ± 31 sec. Histological examination indicated that the thrombi consisted mainly of platelets. The quantitative nature of the transillumination technique was demonstrated by simultaneous measurement of the incorporation of 111In labeled platelets into the thrombus. Thrombus formation, studied at 30 min interval in both femoral veins, showed satisfactory reproducibility overall and within a given animalWith this method we were able to induce a thrombus using a clinically relevant injury and to monitor continuously and reproducibly the kinetics of thrombus formation in a vessel of clinically and surgically relevant size

Persistence of giants: population dynamics of the limpet Scutellastra laticostata on rocky shores in Western Australia

2020 ◽  
Vol 646 ◽  
pp. 79-92
Author(s):  
RE Scheibling ◽  
R Black
Keyword(s):  
Population Dynamics ◽  
Western Australia ◽  
Size Structure ◽  
Survival Rates ◽  
Maximum Size ◽  
High Rate ◽  
Adult Survival ◽  
Rocky Shores ◽  
Adult Size ◽  
Decadal Time Scale

Population dynamics and life history traits of the ‘giant’ limpet Scutellastra laticostata on intertidal limestone platforms at Rottnest Island, Western Australia, were recorded by interannual (January/February) monitoring of limpet density and size structure, and relocation of marked individuals, at 3 locations over periods of 13-16 yr between 1993 and 2020. Limpet densities ranged from 4 to 9 ind. m-2 on wave-swept seaward margins of platforms at 2 locations and on a rocky notch at the landward margin of the platform at a third. Juvenile recruits (25-55 mm shell length) were present each year, usually at low densities (<1 m-2), but localized pulses of recruitment occurred in some years. Annual survival rates of marked limpets varied among sites and cohorts, ranging from 0.42 yr-1 at the notch to 0.79 and 0.87 yr-1 on the platforms. A mass mortality of limpets on the platforms occurred in 2003, likely mediated by thermal stress during daytime low tides, coincident with high air temperatures and calm seas. Juveniles grew rapidly to adult size within 2 yr. Asymptotic size (L∞, von Bertalanffy growth model) ranged from 89 to 97 mm, and maximum size from 100 to 113 mm, on platforms. Growth rate and maximum size were lower on the notch. Our empirical observations and simulation models suggest that these populations are relatively stable on a decadal time scale. The frequency and magnitude of recruitment pulses and high rate of adult survival provide considerable inertia, enabling persistence of these populations in the face of sporadic climatic extremes.

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