scholarly journals The Edit Distance Function and Symmetrization

10.37236/2262 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Ryan R. Martin
Keyword(s):  
Distance Function ◽  
Edit Distance ◽  
Symmetric Difference ◽  
Hereditary Property ◽  
Vertex Set ◽  
Hereditary Properties

The edit distance between two graphs on the same labeled vertex set is the size of the symmetric difference of the edge sets.  The distance between a graph, G, and a hereditary property, ℋ, is the minimum of the distance between G and each G'∈ℋ.  The edit distance function of ℋ is a function of p∈[0,1] and is the limit of the maximum normalized distance between a graph of density p and ℋ.This paper utilizes a method due to Sidorenko [Combinatorica 13(1), pp. 109-120], called "symmetrization", for computing the edit distance function of various hereditary properties.  For any graph H, Forb(H) denotes the property of not having an induced copy of H.  This paper gives some results regarding estimation of the function for an arbitrary hereditary property. This paper also gives the edit distance function for Forb(H), where H is a cycle on 9 or fewer vertices.

scholarly journals Edit Distance and its Computation

10.37236/744 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
József Balogh ◽  
Ryan Martin
Keyword(s):  
Bipartite Graph ◽  
Edit Distance ◽  
Complete Bipartite Graph ◽  
New Method ◽  
Independent Interest ◽  
Regularity Lemma ◽  
Hereditary Property ◽  
Vertex Set ◽  
Hereditary Properties ◽  
Szemeredi's Regularity Lemma

In this paper, we provide a method for determining the asymptotic value of the maximum edit distance from a given hereditary property. This method permits the edit distance to be computed without using Szemerédi's Regularity Lemma directly. Using this new method, we are able to compute the edit distance from hereditary properties for which it was previously unknown. For some graphs $H$, the edit distance from ${\rm Forb}(H)$ is computed, where ${\rm Forb}(H)$ is the class of graphs which contain no induced copy of graph $H$. Those graphs for which we determine the edit distance asymptotically are $H=K_a+E_b$, an $a$-clique with $b$ isolated vertices, and $H=K_{3,3}$, a complete bipartite graph. We also provide a graph, the first such construction, for which the edit distance cannot be determined just by considering partitions of the vertex set into cliques and cocliques. In the process, we develop weighted generalizations of Turán's theorem, which may be of independent interest.

scholarly journals A Note on the Speed of Hereditary Graph Properties

10.37236/644 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vadim V. Lozin ◽  
Colin Mayhill ◽  
Victor Zamaraev
Keyword(s):  
Practical Importance ◽  
Bipartite Graphs ◽  
Hereditary Property ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Graph Properties ◽  
Graph Property ◽  
Vertex Set ◽  
Split Graphs ◽  
Hereditary Properties

For a graph property $X$, let $X_n$ be the number of graphs with vertex set $\{1,\ldots,n\}$ having property $X$, also known as the speed of $X$. A property $X$ is called factorial if $X$ is hereditary (i.e. closed under taking induced subgraphs) and $n^{c_1n}\le X_n\le n^{c_2n}$ for some positive constants $c_1$ and $c_2$. Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored, although this family includes many properties of theoretical or practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial properties, we propose the following conjecture: the speed of a hereditary property $X$ is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in $X$, co-bipartite graphs in $X$ and split graphs in $X$. In this note, we verify the conjecture for hereditary properties defined by forbidden induced subgraphs with at most 4 vertices.

scholarly journals On the edit distance function of the random graph

2021 ◽  
pp. 1-23
Author(s):  
Ryan R. Martin ◽  
Alex W. N. Riasanovsky
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Distance Function ◽  
Edit Distance ◽  
Golden Ratio ◽  
Edge Density ◽  
Hereditary Property ◽  
Maximum Proportion ◽  
Speed Function ◽  
Vertex Disjoint

Abstract Given a hereditary property of graphs $\mathcal{H}$ and a $p\in [0,1]$ , the edit distance function $\textrm{ed}_{\mathcal{H}}(p)$ is asymptotically the maximum proportion of edge additions plus edge deletions applied to a graph of edge density p sufficient to ensure that the resulting graph satisfies $\mathcal{H}$ . The edit distance function is directly related to other well-studied quantities such as the speed function for $\mathcal{H}$ and the $\mathcal{H}$ -chromatic number of a random graph. Let $\mathcal{H}$ be the property of forbidding an Erdős–Rényi random graph $F\sim \mathbb{G}(n_0,p_0)$ , and let $\varphi$ represent the golden ratio. In this paper, we show that if $p_0\in [1-1/\varphi,1/\varphi]$ , then a.a.s. as $n_0\to\infty$ , \begin{align*} {\textrm{ed}}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0} \cdot\min\left\{ \frac{p}{-\log(1-p_0)}, \frac{1-p}{-\log p_0} \right\}. \end{align*} Moreover, this holds for $p\in [1/3,2/3]$ for any $p_0\in (0,1)$ . A primary tool in the proof is the categorization of p-core coloured regularity graphs in the range $p\in[1-1/\varphi,1/\varphi]$ . Such coloured regularity graphs must have the property that the non-grey edges form vertex-disjoint cliques.

Parallel Residues

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Distance Function ◽  
Length Function ◽  
Coxeter System ◽  
Vertex Set ◽  
Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

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.

Autometrization and the Symmetric Difference

1953 ◽  
Vol 5 ◽  
pp. 324-331 ◽  
Author(s):  
J. G. Elliott
Keyword(s):  
Boolean Algebra ◽  
Distance Function ◽  
Symmetric Difference ◽  
Group Operation ◽  
Metric Distance

The fact that the symmetric difference is a group operation in a Boolean algebra is, of course, well known. Not so well known is the fact observed by Ellis [3] that it possesses some of the desirable properties of a metric distance function. Specifically, if * denotes this operation, it is easy to verify that

Research on the Hereditary Properties of the Cartesian Product Operation of Graphs

2012 ◽  
Vol 241-244 ◽  
pp. 2802-2806
Author(s):  
Hua Dong Wang ◽  
Bin Wang ◽  
Yan Zhong Hu
Keyword(s):  
Euler Characteristic ◽  
Cartesian Product ◽  
Hamilton Cycle ◽  
Hereditary Property ◽  
Constant Property ◽  
Product Graph ◽  
Product Operation ◽  
Graph Operation ◽  
Hereditary Properties

This paper defined the hereditary property (or constant property) concerning graph operation, and discussed various forms of the hereditary property under the circumstance of Cartesian product graph operation. The main conclusions include: The non-planarity and Hamiltonicity of graph are hereditary concerning the Cartesian product, but planarity of graph is not, Euler characteristic and non-hamiltonicity of graph are not hereditary as well. Therefore, when we applied this principle into practice, we testified that Hamilton cycle does exist in hypercube.

scholarly journals The edit distance function of some graphs

2020 ◽  
Vol 40 (3) ◽  
pp. 807
Author(s):  
Yumei Hu ◽  
Yongtang Shi ◽  
Yarong Wei
Keyword(s):  
Distance Function ◽  
Edit Distance

scholarly journals Some Extremal Problems for Hereditary Properties of Graphs

10.37236/3419 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Vladimir Nikiforov
Keyword(s):  
Real Number ◽  
Extremal Problems ◽  
Hereditary Property ◽  
Hereditary Properties ◽  
Extremal Parameter ◽  
Stone Theorem

Given an infinite hereditary property of graphs $\mathcal{P}$, the principal extremal parameter of $\mathcal{P}$ is the value\[ \pi\left( \mathcal{P}\right) =\lim_{n\rightarrow\infty}\binom{n}{2}^{-1}\max\{e\left( G\right) :\text{ }G\in\mathcal{P}\text{ and }v\left(G\right) =n\}.\]The Erdős-Stone theorem gives $\pi\left( \mathcal{P}\right) $ if $\mathcal{P}$ is monotone, but this result does not apply to hereditary $\mathcal{P}$. Thus, one of the results of this note is to establish $\pi\left( \mathcal{P}\right) $ for any hereditary property $\mathcal{P}.$Similar questions are studied for the parameter $\lambda^{\left( p\right)}\left( G\right)$, defined for every real number $p\geq1$ and every graph $G$ of order $n$ as\[\lambda^{\left( p\right) }\left( G\right) =\max_{\left\vert x_{1}\right\vert^{p}\text{ }+\text{ }\cdots\text{ }+\text{ }\left\vert x_{n}\right\vert ^{p} \text{ }=\text{ }1}2\sum_{\{u,v\}\in E\left( G\right) }x_{u}x_{v}.\]It is shown that the limit\[ \lambda^{\left( p\right) }\left( \mathcal{P}\right) =\lim_{n\rightarrow\infty}n^{2/p-2}\max\{\lambda^{\left( p\right) }\left( G\right) :\text{ }G\in \mathcal{P}\text{ and }v\left( G\right) =n\}\]exists for every hereditary property $\mathcal{P}$.A key result of the note is the equality \[\lambda^{(p)}\left( \mathcal{P}\right) =\pi\left( \mathcal{P}\right) ,\]which holds for all $p>1.$ In particular, edge extremal problems andspectral extremal problems for graphs are asymptotically equivalent.

Sign in / Sign up

Export Citation Format

Share Document