scholarly journals Spaces for which the diagonal has a closed neighborhood base

1987 ◽  
Vol 53 (1) ◽  
pp. 49-56 ◽  
Author(s):  
Klaas Pieter Hart
Keyword(s):  
Neighborhood Base ◽  
Closed Neighborhood

scholarly journals The Metrizability of Spaces whose Diagonals have a Countable Base

1977 ◽  
Vol 20 (4) ◽  
pp. 513-514 ◽  
Author(s):  
John Ginsburg
Keyword(s):  
Metrizable Space ◽  
Countable Base ◽  
Neighborhood Base ◽  
Isolated Points

AbstractIt is shown that the diagonal of X has a countable neighborhood base in X × X if and only if X is a metrizable space whose set of non-isolated points is compact.

scholarly journals Lower Bounds for Identifying Codes in some Infinite Grids

10.37236/394 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ryan Martin ◽  
Brendon Stanton
Keyword(s):  
Lower Bounds ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Locally Finite ◽  
Identifying Codes ◽  
Identifying Code ◽  
Hexagonal Grids ◽  
Closed Neighborhood ◽  
Definition Of ◽  
Infinite Grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

Majority Roman domination in graphs

2020 ◽  
pp. 2150062
Author(s):  
S. Anandha Prabhavathy
Keyword(s):  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Closed Neighborhood ◽  
Domination In Graphs ◽  
Dominating Function

A Majority Roman Dominating Function (MRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) the sum of its function values over at least half the closed neighborhood is at least one and (ii) every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a MRDF is the sum of its function values over all vertices. The Majority Roman Domination Number of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text]. In this paper, we initiate the study of Majority Roman Domination in Graphs.

scholarly journals The closed neighborhood and filter conditions in solid sequence spaces

1989 ◽  
Vol 12 (4) ◽  
pp. 625-632
Author(s):  
P. D. Johnson, Jr.
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Sequence Spaces ◽  
Schauder Basis ◽  
Normed Spaces ◽  
Neighborhood Conditions ◽  
Neighborhood Condition ◽  
Closed Neighborhood

Let E be a topological vector space of scalar sequences, with topologyτ;(E,τ)satisfies the closed neighborhood condition iff there is a basis of neighborhoods at the origin, forτ, consisting of sets whlch are closed with respect to the topologyπof coordinate-wise convergence on E;(E,τ)satisfies the filter condition iff every filter, Cauchy with respect toτ, convergent with respect toπ, converges with respect toτ.Examples are given of solid (definition below) normed spaces of sequences which (a) fail to satisfy the filter condition, or (b) satisfy the filter condition, but not the closed neighborhood condition. (Robertson and others have given examples fulfilling (a), and examples fulfilling (b), but these examples were not solid, normed sequence spaces.) However, it is shown that among separated, separable solid pairs(E,τ), the filter and closed neighborhood conditions are equivalent, and equivalent to the usual coordinate sequences constituting an unconditional Schauder basis for(E,τ). Consequently, the usual coordinate sequences do constitute an unconditional Schauder basis in every complete, separable, separated, solid pair(E,τ).

scholarly journals Closed neighborhood ideal of a graph

2020 ◽  
Vol 50 (3) ◽  
pp. 1097-1107
Author(s):  
Leila Sharifan ◽  
Somayeh Moradi
Keyword(s):  
Closed Neighborhood

scholarly journals Signed strong Roman domination in graphs

2017 ◽  
Vol 48 (2) ◽  
pp. 135-147 ◽  
Author(s):  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar ◽  
Leila Asgharsharghi
Keyword(s):  
Maximum Degree ◽  
Domination Number ◽  
Simple Graph ◽  
Roman Domination ◽  
Sharp Bounds ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Closed Neighborhood ◽  
Domination In Graphs ◽  
Dominating Function

Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximum degree $\Delta$. A signed strong Roman dominating function (abbreviated SStRDF) on a graph $G$ is a function $f:V\to \{-1,1,2,\ldots,\lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the conditions that (i) for every vertex $v$ of $G$, $\sum_{u\in N[v]} f(u)\ge 1$, where $N[v]$ is the closed neighborhood of $v$ and (ii) every vertex $v$ for which $f(v)=-1$ is adjacent to at least one vertex $u$ for which $f(u)\ge 1+\lceil\frac{1}{2}|N(u)\cap V_{-1}|\rceil$, where $V_{-1}=\{v\in V \mid f(v)=-1\}$. The minimum of the values $\sum_{v\in V} f(v)$, taken over all signed strong Roman dominating functions $f$ of $G$, is called the signed strong Roman domination number of $G$ and is denoted by $\gamma_{ssR}(G)$. In this paper we initiate the study of the signed strong Roman domination in graphs and present some (sharp) bounds for this parameter.

scholarly journals Local chromatic number and topology

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gábor Simonyi ◽  
Gábor Tardos
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Topological Method ◽  
Proper Coloring ◽  
And Topology ◽  
Minimum Number ◽  
International Audience ◽  
Closed Neighborhood

International audience The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of $4$-chromatic graphs and, in particular, on surface quadrangulations.

scholarly journals Free paratopological groups

2015 ◽  
Vol 16 (2) ◽  
pp. 89
Author(s):  
Ali Sayed Elfard
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Inductive Limit ◽  
Infinite Cardinal ◽  
Alexandroff Space ◽  
Limit Property ◽  
Neighborhood Base ◽  
Paratopological Group

Let FP(X) be the free paratopological group on a topological space X in the sense of Markov. In this paper, we study the group FP(X) on a $P_\alpha$-space $X$ where $\alpha$ is an infinite cardinal and then we prove that the group FP(X) is an Alexandroff space if X is an Alexandroff space. Moreover, we introduce a~neighborhood base at the identity of the group FP(X) when the space X is Alexandroff and then we give some properties of this neighborhood base. As applications of these, we prove that the group FP(X) is T_0 if X is T_0, we characterize the spaces X for which the group FP(X) is a topological group and then we give a class of spaces $X$ for which the group FP(X) has the inductive limit property.

Symmetrizable, -, and Weakly First Countable Spaces

1977 ◽  
Vol 29 (3) ◽  
pp. 480-488 ◽  
Author(s):  
R. M. Stephenson
Keyword(s):  
Hausdorff Space ◽  
Cardinal Number ◽  
Locally Compact ◽  
Neighborhood Base

A number of results are given concerning the character and cardinality of symmetrizable and related spaces. An example is given of a symmetrizable Hausdorff space containing a point that is not a regular Gδ , and a proof is given that if a point p of a symmetrizable Hausdorff space has a neighborhood base of cardinality , then p is a Gδ . It is shown that for each cardinal number m there exists a locally compact, pseudocompact, Hausdorff -space X with |X| ≧ m. Several questions of A. V. Arhangel'skii and E. Michael are partially answered.

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