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 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 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 Graph Operations and Neighbor Rupture Degree

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Saadet Kandİlcİ ◽  
Goksen Bacak-Turan ◽  
Refet Polat
Keyword(s):  
Communication Network ◽  
Connected Components ◽  
Maximum Order ◽  
Graph Operations ◽  
Communication Links ◽  
Closed Neighborhood

In a communication network, the vulnerability parameters measure the resistance of the network to disruption of operation after the failure of certain stations or communication links. A vertex subversion strategy of a graph , say , is a set of vertices in whose closed neighborhood is removed from . The survival subgraph is denoted by . The neighbor rupture degree of , , is defined to be , where is any vertex subversion strategy of , is the number of connected components in and is the maximum order of the components of (G. Bacak Turan, 2010). In this paper we give some results for the neighbor rupture degree of the graphs obtained by some graph operations.

scholarly journals Solutions to four open problems on quorum colorings of graphs

2021 ◽  
Author(s):  
Rafik Sahbi
Keyword(s):  
Maximum Cardinality ◽  
Open Problems ◽  
Coloring Number ◽  
Vertex Set ◽  
Good Characterization ◽  
Closed Neighborhood

A partition $\pi=\{V_{1},V_{2},...,V_{k}\}$ of the vertex set $V$ of a graph $G$ into $k$ color classes $V_{i},$ with $1\leq i\leq k$ is called a quorum coloring if for every vertex $v\in V,$ at least half of the vertices in the closed neighborhood $N[v]$ of $v$ have the same color as $v$. The maximum cardinality of a quorum coloring of $G$ is called the quorum coloring number of $G$ and is denoted $\psi_{q}(G).$ In this paper, we give answers to four open problems stated in 2013 by Hedetniemi, Hedetniemi, Laskar and Mulder. In particular, we show that there is no good characterization of the graphs $G$ with $\psi_{q}(G)=1$ nor for those with $\psi_{q} (G)>1$ unless $\mathcal{P}\neq\mathcal{NP}\cap co-\mathcal{NP}.$ We also construct several new infinite  families of such graphs, one of which the diameter $diam(G)$ of $G$ is not bounded.

ON LOCALLY-BALANCED 2-PARTITIONS OF SOME CLASSES OF GRAPHS

2020 ◽  
Vol 54 (1 (251)) ◽  
pp. 9-19
Author(s):  
A.H. Gharibyan
Keyword(s):  
Polynomial Time ◽  
Necessary Conditions ◽  
Polynomial Time Algorithms ◽  
Closed Neighborhood

In this paper we obtain some conditions for the existence of locally-balanced 2-partitions with an open (with a closed) neighborhood of some classes of graphs. In particular, we give necessary conditions for the existence of locallybalanced 2-partitions of even and odd graphs. We also obtain some results on the existence of locally-balanced 2-partitions of rook’s graphs and powers of cycles. In particular, we prove that if \(m,n \geq 2\), then the graph \(K_m \Box K_n\) has a locally-balanced 2-partition with a closed neighborhood if and only if m and n are even. Moreover, all our proofs are constructive and provide polynomial time algorithms for constructing the required 2-partitions.

The signed Roman domination number of two classes graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050024
Author(s):  
Xia Hong ◽  
Tianhu Yu ◽  
Zhengbang Zha ◽  
Huihui Zhang
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Double Star ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Closed Neighborhood

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. A signed Roman dominating function (SRDF) of [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) [Formula: see text] for each [Formula: see text], where [Formula: see text] is the set, called closed neighborhood of [Formula: see text], consists of [Formula: see text] and the vertex of [Formula: see text] adjacent to [Formula: see text] (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 SRDF [Formula: see text] is [Formula: see text]. The signed Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of a SRDF of [Formula: see text]. In this paper, we determine the exact values of signed Roman domination number of spider and double star. Specially, one of them generalizes the known result.

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