Testing hereditary properties efficiently on average

2005 ◽  
pp. 100-116 ◽  
Author(s):  
Jens Gustedt ◽  
Angelika Steger
Keyword(s):  
Hereditary Properties

Minimal Characteristic Algebras for Rectangular k-Normal Identities

2011 ◽  
Vol 18 (04) ◽  
pp. 611-628
Author(s):  
K. Hambrook ◽  
S. L. Wismath
Keyword(s):  
Hereditary Property ◽  
Fixed Type ◽  
Hereditary Properties

A characteristic algebra for a hereditary property of identities of a fixed type τ is an algebra [Formula: see text] such that for any variety V of type τ, we have [Formula: see text] if and only if every identity satisfied by V has the property p. This is equivalent to [Formula: see text] being a generator for the variety determined by all identities of type τ which have property p. Płonka has produced minimal (smallest cardinality) characteristic algebras for a number of hereditary properties, including regularity, normality, uniformity, biregularity, right- and leftmost, outermost, and external-compatibility. In this paper, we use a construction of Płonka to study minimal characteristic algebras for the property of rectangular k-normality. In particular, we construct minimal characteristic algebras of type (2) for k-normality and rectangularity for 1 ≤ k ≤ 3.

scholarly journals On Growth Rates of Permutations, Set Partitions, Ordered Graphs and Other Objects

10.37236/799 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Martin Klazar
Keyword(s):  
Growth Rates ◽  
Fibonacci Numbers ◽  
Complete Graphs ◽  
Linear Orders ◽  
Set Partitions ◽  
Ordered Graphs ◽  
Containment Order ◽  
Hereditary Properties ◽  
Finite Linear ◽  
Lower Ideal

For classes ${\cal O}$ of structures on finite linear orders (permutations, ordered graphs etc.) endowed with containment order $\preceq$ (containment of permutations, subgraph relation etc.), we investigate restrictions on the function $f(n)$ counting objects with size $n$ in a lower ideal in $({\cal O},\preceq)$. We present a framework of edge $P$-colored complete graphs $({\cal C}(P),\preceq)$ which includes many of these situations, and we prove for it two such restrictions (jumps in growth): $f(n)$ is eventually constant or $f(n)\ge n$ for all $n\ge 1$; $f(n)\le n^c$ for all $n\ge 1$ for a constant $c>0$ or $f(n)\ge F_n$ for all $n\ge 1$, $F_n$ being the Fibonacci numbers. This generalizes a fragment of a more detailed theorem of Balogh, Bollobás and Morris on hereditary properties of ordered graphs.

scholarly journals Clique-Width and the Speed of Hereditary Properties

10.37236/124 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter Allen ◽  
Vadim Lozin ◽  
Michaël Rao
Keyword(s):  
Hereditary Class ◽  
Bell Number ◽  
Hereditary Properties ◽  
The Relationship

In this paper, we study the relationship between the number of $n$-vertex graphs in a hereditary class $\cal X$, also known as the speed of the class $\cal X$, and boundedness of the clique-width in this class. We show that if the speed of $\cal X$ is faster than $n!c^n$ for any $c$, then the clique-width of graphs in $\cal X$ is unbounded, while if the speed does not exceed the Bell number $B_n$, then the clique-width is bounded by a constant. The situation in the range between these two extremes is more complicated. This area contains both classes of bounded and unbounded clique-width. Moreover, we show that classes of graphs of unbounded clique-width may have slower speed than classes where the clique-width is bounded.

The Symmetric Rank One Formula and its Application in Discrete Nonlinear Optimization

1988 ◽  
Author(s):  
J. Z. Cha ◽  
R. W. Mayne
Keyword(s):  
Hessian Matrix ◽  
Attractive Feature ◽  
Recursive Quadratic Programming ◽  
Rank One ◽  
Hereditary Properties ◽  
Symmetric Rank One ◽  
Derivative Information ◽  
Theoretical Considerations ◽  
Specific Search ◽  
Search Directions

Abstract The hereditary properties of the Symmetric Rank One (SRI) update formula for numerically accumulating second order derivative information are studied. The unique advantage of the SR1 formula is that it does not require specific search directions for development of the Hessian matrix. This is an attractive feature for optimization applications where arbitrary search directions may be necessary. This paper explores the use of the SR1 formula within a procedure based on recursive quadratic programming (RQP) for solving a class of mixed discrete constrained nonlinear programming (MDCNP) problems. Theoretical considerations are presented along with numerical examples which illustrate the procedure and the utility of SR1.

Hereditary properties of some special spaces

1961 ◽  
Vol 12 (1) ◽  
pp. 129-133 ◽  
Author(s):  
Carl W. Kohls
Keyword(s):  
Hereditary Properties
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