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Minimal dominating sets in graph classes: Combinatorial bounds and enumeration

2013
*
Theoretical Computer Science
*

There is a

doi:10.1016/j.tcs.2013.03.026
fatcat:ivmx7hhl4fh6xf6ew6cdjis5ya
*graph*on n vertices with 15 n/6*minimal**dominating**sets*. This gives a lower bound*of*1.5704 n*for*the*maximum**number**of**minimal**dominating**sets*. ... This gives a lower bound*of*1.5704 n*for*the*maximum**number**of**minimal**dominating**sets*.*Graph*classes Our work is dealing with some well-known*graph*classes. ...##
###
Enumerating minimal connected dominating sets in graphs of bounded chordality

2016
*
Theoretical Computer Science
*

We establish

doi:10.1016/j.tcs.2016.03.026
fatcat:eucvh6r4p5gl7cpvymf7qmuugu
*enumeration*algorithms as well as lower*and*upper bounds*for*the*maximum**number**of**minimal*connected*dominating**sets*in such*graphs*. ... In particular, we present algorithms to*enumerate*all*minimal*connected*dominating**sets**of**chordal**graphs*in time O(1.7159 n ),*of*split*graphs*in time O(1.3803 n ),*and**of*AT-free, strongly*chordal*,*and*... In this paper we initiate the study*of*the*enumeration**and**maximum**number**of**minimal*connected*dominating**sets*in a given*graph*. ...##
###
Enumeration of Enumeration Algorithms
[article]

2016
*
arXiv
*
pre-print

In this paper, we

arXiv:1605.05102v2
fatcat:bwmhesagkje5hmynqyij7jv224
*enumerate**enumeration*problems*and*algorithms. This survey is under construction. If you know some results not in this survey or there is anything wrong, please let me know. ... Reference [115] 2.12.15*Enumeration**of*all*minimal**dominating**sets*in a*chordal*bipartite*graph*Input A*chordal*bipartite*graph*G. Output All*minimal**dominating**sets*in G. ... Reference [140] 2.12.14*Enumeration**of*all*minimal**dominating**sets*in a P 6 -free*chordal**graph*Input A P 6 -free*chordal**graph*G = (V, E). Output All*minimal**dominating**sets*in G. ...##
###
A Note on the Maximum Number of Minimal Connected Dominating Sets in a Graph
[article]

2021
*
arXiv
*
pre-print

This improves the previously known lower bound

arXiv:2111.06026v1
fatcat:tgatzl4b55eynhhkrhlackckya
*of*Ω(1.4422^n)*and*reduces the gap between lower*and*upper bounds*for*input-sensitive*enumeration**of**minimal*connected*dominating**sets*in general*graphs*as ... We prove constructively that the*maximum*possible*number**of**minimal*connected*dominating**sets*in a connected undirected*graph**of*order n is in Ω(1.489^n). ... In this note we report an improved lower bound on the*maximum**number**of**minimal*connected*dominating**sets*in a*graph*. ...##
###
A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs
[article]

2014
*
arXiv
*
pre-print

We give a polynomial delay algorithm to list the

arXiv:1407.2036v1
fatcat:wieaa7sh3zc7tjusqvvhs6ei34
*set**of**minimal**dominating**sets*in*chordal**graphs*, an important*and*well-studied*graph*class where such an algorithm was open*for*a while. ... An output-polynomial algorithm*for*the listing*of**minimal**dominating**sets*in*graphs*is a challenging open problem*and*is known to be equivalent to the well-known Transversal problem which asks*for*an output-polynomial ...*For*the*enumeration**of**minimal**dominating**sets*in*chordal**graphs*the simplest strategy consists in following this ordering as follows. ...##
###
A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs
[chapter]

2016
*
Lecture Notes in Computer Science
*

We give a polynomial delay algorithm to list the

doi:10.1007/978-3-662-53174-7_11
fatcat:553be4juyjh67ew7zs7d5hnw44
*set**of**minimal**dominating**sets*in*chordal**graphs*, an important*and*well-studied*graph*class where such an algorithm was open*for*a while. ... An output-polynomial algorithm*for*the listing*of**minimal**dominating**sets*in*graphs*is a challenging open problem*and*is known to be equivalent to the well-known Transversal problem which asks*for*an output-polynomial ...*For*the*enumeration**of**minimal**dominating**sets*in*chordal**graphs*the simplest strategy consists in following this ordering as follows. ...##
###
Minimal Dominating Sets in Graph Classes: Combinatorial Bounds and Enumeration
[chapter]

2012
*
Lecture Notes in Computer Science
*

*For*several classes

*of*

*graphs*, we substantially improve the upper bound on the

*maximum*

*number*

*of*

*minimal*

*dominating*

*sets*in

*graphs*on n vertices. ...

*For*all considered

*graph*classes, the upper bound proofs are constructive

*and*can easily be transformed into algorithms

*enumerating*all

*minimal*

*dominating*

*sets*

*of*the input

*graph*. ... [11] showed that all

*minimal*

*dominating*

*sets*in a split

*graph*G can be

*enumerated*in time polynomial in the

*number*

*of*

*minimal*

*dominating*

*sets*

*of*G. ...

##
###
On the maximum number of minimal connected dominating sets in convex bipartite graphs
[article]

2019
*
arXiv
*
pre-print

Our algorithm implies a corresponding upper bound

arXiv:1908.02174v1
fatcat:q353hmpvovg5jeiqjbn7cdsdiu
*for*the*number**of**minimal*connected*dominating**sets**for*this*graph*class. ... The*enumeration**of**minimal*connected*dominating**sets*is known to be notoriously hard*for*general*graphs*. ... In this paper we study the*enumeration**and**maximum**number**of**minimal*connected*dominating**sets*in convex bipartite*graphs*,*and*we prove that the*number**of**minimal*connected*dominating**sets*in a convex ...##
###
Enumeration and maximum number of minimal connected vertex covers in graphs

2018
*
European journal of combinatorics (Print)
*

*For*

*graphs*

*of*bounded

*chordality*, we are able to give a better upper bound,

*and*

*for*

*chordal*

*graphs*

*and*distance-hereditary

*graphs*we are able to give tight bounds on the

*maximum*

*number*

*of*

*minimal*connected ... In this paper we show that the

*maximum*

*number*

*of*

*minimal*connected vertex covers

*of*a

*graph*is O(1.8668 n ),

*and*these can be

*enumerated*in time O(1.8668 n ). ... Examples

*of*such recent results, both on general

*graphs*

*and*on some

*graph*classes, concern the

*enumeration*

*and*

*maximum*

*number*

*of*

*minimal*

*dominating*

*sets*,

*minimal*feedback vertex

*sets*,

*minimal*subset feedback ...

##
###
Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs
[article]

2016
*
arXiv
*
pre-print

*For*

*graphs*

*of*

*chordality*at most 5, we are able to give a better upper bound,

*and*

*for*

*chordal*

*graphs*

*and*distance-hereditary

*graphs*we are able to give tight bounds on the

*maximum*

*number*

*of*

*minimal*connected ... In this paper we show that the

*maximum*

*number*

*of*

*minimal*connected vertex covers

*of*a

*graph*is at most 1.8668^n,

*and*these can be

*enumerated*in time O(1.8668^n). ... Examples

*of*such recent results, both on general

*graphs*

*and*on some

*graph*classes, concern the

*enumeration*

*and*

*maximum*

*number*

*of*

*minimal*

*dominating*

*sets*,

*minimal*feedback vertex

*sets*,

*minimal*subset feedback ...

##
###
Subset feedback vertex sets in chordal graphs

2014
*
Journal of Discrete Algorithms
*

We give an algorithm with running time O(1.6708 n ) that

doi:10.1016/j.jda.2013.09.005
fatcat:m74entnpo5e6vbcv55gmiudjxm
*enumerates*all*minimal*subset feedback vertex*sets*on*chordal**graphs*on n vertices. ... We also obtain that a*chordal**graph*G has at most 1.6708 n*minimal*subset feedback vertex*sets*, regardless*of*S . ... More recently, the*maximum**numbers**and**enumeration**of*objects like*minimal**dominating**sets*,*minimal*feedback vertex*sets*,*minimal*subset feedback vertex*sets*,*minimal*separators,*and*potential maximal ...##
###
On the Neighbourhood Helly of Some Graph Classes and Applications to the Enumeration of Minimal Dominating Sets
[chapter]

2012
*
Lecture Notes in Computer Science
*

As a consequence, we obtain output-polynomial time algorithms

doi:10.1007/978-3-642-35261-4_32
fatcat:fkjgolto4rb75pxadir2i34ucy
*for**enumerating*the*set**of**minimal**dominating**sets**of*line*graphs**and*path*graphs*. ... Therefore, there exists an output-polynomial time algorithm that*enumerates*the*set**of**minimal*edge-*dominating**sets**of*any*graph*. ... We denote by D(G) the*set**of*(inclusionwise)*minimal**dominating**sets**of*a*graph*G. ...##
###
On Distance-d Independent Set and other problems in graphs with few minimal separators
[article]

2016
*
arXiv
*
pre-print

We also provide polynomial algorithms

arXiv:1607.04545v1
fatcat:s3y6oshzzfhhdbfnufiyv43ide
*for*Connected Vertex Cover*and*Connected Feedback Vertex*Set*on subclasses*of*including*chordal**and*circular-arc*graphs*,*and*we discuss variants*of*independent*domination*... Fomin*and*Villanger (STACS 2010) proved that*Maximum*Independent*Set*, Feedback Vertex*Set*,*and*more generally the problem*of*finding a*maximum*induced subgraph*of*treewith at most a constant t, can be ... We thank Iyad Kanj*for*fruitful discussions on the subject. ...##
###
Counting the number of independent sets in chordal graphs

2008
*
Journal of Discrete Algorithms
*

We study some counting

doi:10.1016/j.jda.2006.07.006
fatcat:pt3o77gc5zcrhdirjtbnrx5kwe
*and**enumeration*problems*for**chordal**graphs*, especially concerning independent*sets*. ... With similar ideas, we show that*enumeration*(namely, listing)*of*the independent*sets*, the*maximum*independent*sets*,*and*the independent*sets**of*a fixed size in a*chordal**graph*can be done in constant ... Acknowledgement The authors thank Masashi Kiyomi*for*enlightening discussions*and*pointing out the work by Chang [7] . The authors are grateful to L. Shankar Ram*for*pointing out a paper [5] . ...##
###
Weighted domination of independent sets
[article]

2017
*
arXiv
*
pre-print

The independent

arXiv:1709.09889v1
fatcat:rgfu3beb4jeazeyaccs6ecbtsu
*domination**number*γ^i(G)*of*a*graph*G is the*maximum*, over all independent*sets*I,*of*the*minimal**number**of*vertices needed to*dominate*I. ... It is known abz that in*chordal**graphs*γ^i is equal to γ, the ordinary*domination**number*. ... We say that f is w-*dominating*if it w-*dominates*V . The independent*domination**number*γ i w (G) is the*maximum*over all independent*sets*I*of*the*minimal*size*of*an integral function w-*dominating*I. ...
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