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Dominating Sets and Connected Dominating Sets in Dynamic Graphs
[article]

2019
*
arXiv
*
pre-print

In this paper we study the dynamic versions of two basic graph problems: Minimum

arXiv:1901.09877v1
fatcat:oveklp63cvccnhnrzo4ujdjdli
*Dominating**Set*and its variant Minimum Connected*Dominating**Set*. ... As a further application of our approach, we also present an algorithm that maintains a Minimal*Dominating**Set*in O(min(Δ, √(m))) per update. ... The algorithm must find a new vertex to*dominate*u: we simply add u to the*dominating**set*. ...##
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Dominating Sets and Connected Dominating Sets in Dynamic Graphs

2019
*
Symposium on Theoretical Aspects of Computer Science
*

In this paper we study the dynamic versions of two basic graph problems: Minimum

doi:10.4230/lipics.stacs.2019.35
dblp:conf/stacs/HjulerIPS19
fatcat:q4rqsmigcjhuxd4wynla2z7f5m
*Dominating**Set*and its variant Minimum Connected*Dominating**Set*. ... As a further application of our approach, we also present an algorithm that maintains a Minimal*Dominating**Set*in O(min(∆, √ m )) per update. ... A O(min(∆, √ m) ) amortized algorithm for Minimal*Dominating**Set*35:15 for some vertex w and OnlyBy(w) = {u}: in other terms, u was*dominated*only by w, and w was in the*dominating**set*only to*dominate*...##
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Connected Dominating Sets
[chapter]

2012
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Ad Hoc and Sensor Wireless Networks: Architectures, Algorithms and Protocols
*

Thus, a Connected

doi:10.2174/978160805018510901010019
fatcat:cixydjydrba4bhvfkya3pryz3e
*Dominating**Set*(CDS) has been recommended to serve as a virtual backbone for a WSN to reduce routing overhead. ... After that, we focus on connected*dominating**set*algorithms, including both centralized and distributed, for how to construct CDS. Theoretical analysis are also presented. ... A*set*C ⊂ V is a k-connected m-*dominating**set*(kmCDS) of graph G(V,E)if the induced subgraph G (C, E ) is k-vertex connected and the*set*C is also an m-*dominating**set*of G. ...##
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Online Dominating Set

2018
*
Algorithmica
*

This paper is devoted to the online

doi:10.1007/s00453-018-0519-1
fatcat:onlesjjc4vdynnftzfrl6gdgua
*dominating**set*problem and its variants. ... There is relatively little difference observed between three of the variants of*Dominating**Set*considered:*Dominating**Set*, Connected*Dominating**Set*, and Total*Dominating**Set*. ... For*Dominating**Set*, Connected*Dominating**Set*, and Total*Dominating**Set*, we start by using the size of a given*dominating**set*to bound the sizes of some connected or incremental equivalents. ...##
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Online Dominating Set
[article]

2018
*
arXiv
*
pre-print

This paper is devoted to the online

arXiv:1604.05172v2
fatcat:6uthyextzndqrbtyxdarsc2cuy
*dominating**set*problem and its variants. ... There is relatively little difference observed between three of the variants of*Dominating**Set*considered:*Dominating**Set*, Connected*Dominating**Set*, and Total*Dominating**Set*. ... For*Dominating**Set*, Connected*Dominating**Set*, and Total*Dominating**Set*, we start by using the size of a given*dominating**set*to bound the sizes of some connected or incremental equivalents. ...##
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Online Dominating Set and Independent Set
[article]

2021
*
arXiv
*
pre-print

For each of these families, we also present lower bounds of the minimum connected

arXiv:2111.07812v1
fatcat:w75orojoonhdbnuepeg6xdq5qu
*dominating**set*problem. ... We prove that the well known online greedy algorithm for*dominating**set*achieves optimal competitive ratio ζ for any graph. ... A*dominating**set*D is said to be an independent*dominating**set*(IDS) if the induced subgraph G[D] is an independent*set*. ...##
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Partial vs. Complete Domination: t-Dominating Set
[chapter]

2007
*
Lecture Notes in Computer Science
*

*SET*can be solved with exponential small error probability in time O((4 + 6α) t · t 6 · n 1 2α +1 ). ... Unbalanced case: • Good coloring: 2 −t • Recursive call: 1 2 • Total failure probability: Balanced case: • Good coloring: 2 −t • Recursive call: 1 2 · 1 2 • Total failure probability: Let 0 < α ≤ 1/25. t -

*DOMINATING*...

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Dominating Sets and Domination Polynomials of Paths

2009
*
International Journal of Mathematics and Mathematical Sciences
*

A

doi:10.1155/2009/542040
fatcat:6xiwohaf5vchdaq4ddeek6vx3q
*setS*⊆Vis a*dominating**set*ofG, if every vertex inV\Sis adjacent to at least one vertex inS. ... Let𝒫nibe the family of all*dominating**sets*of a pathPnwith cardinalityi, and letd(Pn,j)=|𝒫nj|. In this paper, we construct𝒫ni, and obtain a recursive formula ford(Pn,i). ... Corresponding author : alidino.jumani@salu.edu.The*domination**set*D is now union of the Cartesian product of the*sets*and Lemma 4 : 4 for each G−layer the*domination**set*consists of a single vertex, ...##
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Dominating sets and Domination polynomials of Cycles
[article]

2009
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arXiv
*
pre-print

A

arXiv:0905.3268v1
fatcat:fp3aloyolra6vkucb65l2kowt4
*set*S⊂ V is a*dominating**set*of G, if every vertex in V is adjacent to at least one vertex in S. ... Let C_n^i be the family of*dominating**sets*of a cycle C_n with cardinality i, and let d(C_n,i) = | C_n^i. In this paper, we construct C_n^i, and obtain a recursive formula for d(C_n, i). ... A*set*S ⊆ V is a*dominating**set*of G, if every vertex in V \S is adjacent to at least one vertex in S. The*domination*number γ(G) is the minimum cardinality of a*dominating**set*in G. ...##
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Reconfiguration of dominating sets

2015
*
Journal of combinatorial optimization
*

We explore a reconfiguration version of the

doi:10.1007/s10878-015-9947-x
fatcat:iq6rjeongnhephjksvdusv3np4
*dominating**set*problem, where a*dominating**set*in a graph G is a*set*S of vertices such that each vertex is either in S or has a neighbour in S. ... Two*dominating**sets*are adjacent if one can be formed from the other by the addition or deletion of a single vertex. ... A vertex v in a*dominating**set*S is deletable if S \ {v} is also a*dominating**set*of G. ...##
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Restricted-sum-dominant sets
[article]

2017
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arXiv
*
pre-print

The

arXiv:1712.09226v1
fatcat:gtpmkunahfgvzh4bn34kzvby6m
*set*A is called a sum-*dominant*(SD)*set*if |A + A| > |A - A|, and it is called a restricted sum-domonant (RSD)*set*if |A A| > |A - A|. ... We also provide an explicit construction of infinite sequence of RSD*sets*. ... The restricted-sum-*dominant**sets*of integers have been studied recently by D. Penman and M. Wells [11] . ...##
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Enumerating Connected Dominating Sets
[article]

2022
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arXiv
*
pre-print

delay by showing that a polynomial-delay enumeration algorithm for minimal connected

arXiv:2205.00086v1
fatcat:db75f35k2vdlnjthmegktmcst4
*dominating**sets*implies an affirmative algorithmic solution to the Hitting*Set*Transversal problem. ... More precisely, we prove that it is NP-complete to decide, given a graph G and a vertex*set*U, if there exists a minimal connected*dominating**set*D with U⊆ D, even if G is known to be 2-degenerate. ... We can extend*set*-valued functions to*set*arguments; for instance, N G [S] = v∈S N G [v] for a*set*of vertices S; S is a*dominating**set*if N G [S] = V . ...##
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Reconfiguration of Dominating Sets
[article]

2014
*
arXiv
*
pre-print

We explore a reconfiguration version of the

arXiv:1401.5714v2
fatcat:43cvcehofnak5nkp3td2cjwurq
*dominating**set*problem, where a*dominating**set*in a graph G is a*set*S of vertices such that each vertex is either in S or has a neighbour in S. ... Two*dominating**sets*are adjacent if one can be formed from the other by the addition or deletion of a single vertex. ... A vertex v in a*dominating**set*S is deletable if S \ {v} is also a*dominating**set*of G. ...##
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Reconfiguration of Dominating Sets
[chapter]

2014
*
Lecture Notes in Computer Science
*

We explore a reconfiguration version of the

doi:10.1007/978-3-319-08783-2_35
fatcat:xt5bijpazfc4xhxchombuta7me
*dominating**set*problem, where a*dominating**set*in a graph G is a*set*S of vertices such that each vertex is either in S or has a neighbour in S. ... Two*dominating**sets*are adjacent if one can be formed from the other by the addition or deletion of a single vertex. ... A vertex v in a*dominating**set*S is deletable if S \ {v} is also a*dominating**set*of G. ...##
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Unique Efficient Dominating Sets

2020
*
Open Journal of Discrete Mathematics
*

For a graph G, a

doi:10.4236/ojdm.2020.102006
fatcat:i4maxma4mfdhhdvbqahcrhagz4
*set*D is called a unique efficient*dominating**set*of G if it is the only efficient*dominating**set*of G. ... If a*dominating**set*is both independent and perfect, then it is called an efficient*dominating**set*. ... Zhou*dominates*its neighbors and itself when it is in the*dominating**set*. Using this convention, an efficient*dominating**set*is a*dominating**set*that*dominates*every vertex in a graph exactly once. ...
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