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

2019
*
arXiv
*
pre-print

We present an algorithm to enumerate all

arXiv:1908.02174v1
fatcat:q353hmpvovg5jeiqjbn7cdsdiu
*minimal**connected**dominating**sets**of**convex**bipartite**graphs**in*time O(1.7254^n) where n is*the**number**of*vertices*of**the*input*graph*. ... Hence, to*the*best*of*our knowledge, there is no known input-sensitive algorithm for enumerating*minimal**dominating**sets*, or*one**of*their related*sets*,*in**bipartite**graphs*better than that*of*general*graphs*... However,*the*minimum*connected*1:2*On**the**maximum**number**of**minimal**connected**dominating**sets**in**convex**bipartite**graphs**dominating**set*problem is tractable for*convex**bipartite**graphs*and it can be solved ...##
###
Page 2761 of Mathematical Reviews Vol. , Issue 98E
[page]

1998
*
Mathematical Reviews
*

Mynhardt (SA-PRTR; Pretoria)
98e:05070 05C35 Yu, Bo |Yu, Bo*| (3-WTRL-B; Waterloo,

*ON*)*Convexity**of**minimal*total*dominating*functions*in**graphs*. (English summary) J. ... This paper studies*the**convexity**of**minimal*total*dominating*functions (MTDF). An MTDF / is called universal if*convex*combinations*of*f and any other MTDF are*minimal*. ...##
###
Red Domination in Perfect Elimination Bipartite Graphs
[article]

2022
*
arXiv
*
pre-print

We present a tight upper bound

arXiv:2203.10630v1
fatcat:aawxkajsdvh2ndngynrswyez7q
*on**the**number**of*such*sets**in**bipartite**graphs*, and show that we can calculate that*number**in*linear time for*convex**bipartite**graphs*. ...*The*k red*domination*problem for a*bipartite**graph*G=(X,Y,E) is to find a subset D ⊆ X*of*cardinality at most k that*dominates*vertices*of*Y. ...*In**the*section that follows we present a tight upper bound*on**the**number**of*vertices*in*a red*dominating**set**in*a*bipartite**graph*, show how to calculate*the**number**of*such*sets**in*linear time for*convex*...##
###
Page 7219 of Mathematical Reviews Vol. , Issue 96m
[page]

1996
*
Mathematical Reviews
*

Summary: “An edge

*dominating**set*E*of*a*graph*G is a subset*of**the*edge*set**of*G such that every edge not*in*E is adjacent to at least*one*edge*in*E. ...*The*edge*domination**number**of*G is*the*cardinality*of*a minimum edge*dominating**set**of*G. ...##
###
Page 2849 of Mathematical Reviews Vol. , Issue 86g
[page]

1986
*
Mathematical Reviews
*

For a

*graph*G, a subset*of*vertices D is a*dominating**set*if every vertex not*in*D is adjacent to some vertex*in*D.*The**domination**number*, denoted 4(G), is*the*order*of**the*smallest*dominating**set*. ...*The**number**of*d-*convex**sets**in*a*graph*. (Russian) Izv. Akad. Nauk Moldav. SSR Ser. Fiz.-Tekhn. Mat. ...##
###
Page 6632 of Mathematical Reviews Vol. , Issue 92m
[page]

1992
*
Mathematical Reviews
*

*The*k-independence

*number*a;(G)

*of*G is

*the*

*maximum*

*number*

*of*vertices

*of*a k-independent

*set*

*in*G. ... An extremal

*graph*F has

*the*

*maximum*

*number*

*of*edges

*in*a

*graph*feasible for (G, x); its complement F is

*minimal*for (G, x). ...

##
###
Page 1301 of Mathematical Reviews Vol. , Issue 84d
[page]

1984
*
Mathematical Reviews
*

*On*

*the*minimum

*dominating*pair

*number*

*of*a class

*of*

*graphs*. Caribbean J. Math. 1 (1982), no. 2, 73-76. ...

*In*particular, G is

*the*unique

*graph*which has

*the*

*maximum*

*number*

*of*spanning trees among all

*the*

*graphs*with wv vertices and e edges. ...

##
###
On the complexity of the dominating induced matching problem in hereditary classes of graphs

2011
*
Discrete Applied Mathematics
*

*In*

*the*present paper, we identify a critical class for this problem (i.e., a class lying

*on*a "boundary" separating difficult instances

*of*

*the*problem from polynomially solvable

*ones*) and derive a

*number*...

*The*

*dominating*induced matching problem, also known as efficient edge

*domination*, is

*the*problem

*of*determining whether a

*graph*has an induced matching that

*dominates*every edge

*of*

*the*

*graph*. ... Cardoso was supported by

*the*Centre

*of*Research

*on*Optimization and Control from

*the*Fundação para a Ciência e Tecnologia cofinanced by

*the*European Community Fund FEDER/POCI/2010 and

*the*research

*of*Vadim ...

##
###
Introduction

1993
*
Discrete Applied Mathematics
*

It is known that for chordal

doi:10.1016/0166-218x(93)90217-c
fatcat:iklvodekpnho3nyhvmz6acnvuy
*graphs*,*the*size*of*a largest*minimal**dominating**set*,*the*size*of*a*maximum*independent*set*, and*the*size*of*a*maximum*irredundant*set*are all equal. ...*In*their paper, Elmallah and Stewart provide polynomial time algorithms for*the*minimum*dominating**set*problem and*the**maximum*r-independent*set*problem*on**the*family*of*intersection*graphs**of*chords*of*...##
###
Enumeration of Enumeration Algorithms
[article]

2016
*
arXiv
*
pre-print

*In*this paper, we 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 [95] 2.12.10 Enumeration

*of*all

*minimal*edge

*dominating*

*sets*

*in*a

*bipartite*

*graph*Input A

*bipartite*

*graph*G = (V, E). Output All

*minimal*edge

*dominating*

*sets*

*in*G. ...

##
###
Algorithmic aspects of Roman {3}-domination in graphs

2022
*
Reserche operationelle
*

*In*this paper, first we show that

*the*R3DP is NP-complete for chordal

*graphs*, planar

*graphs*and for two subclasses

*of*

*bipartite*

*graphs*namely, star

*convex*

*bipartite*

*graphs*and comb

*convex*

*bipartite*

*graphs*... vertex $u \

*in*G$, where $N_G(u)$ is

*the*

*set*

*of*vertices adjacent to $u$

*in*$G$, is called a \textit{ Roman \{3\}-

*dominating*function} (R3DF)

*of*$G$. ...

*The*

*domination*

*number*is

*the*minimum cardinality

*of*a

*dominating*

*set*

*in*G and is denoted by γ(G) [25] .

*The*concept

*of*Roman

*domination*was introduced

*in*2004 by Cockayne et al.

*in*[1] . ...

##
###
Page 7560 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

*The*authors are interested

*in*determining

*the*

*maximum*possible

*domination*

*number*

*of*a

*graph*(not necessarily

*connected*) with a given

*number*

*of*vertices, say n, and with a specified minimum degree. ... An asymptotic bound is derived for

*the*

*number*

*of*

*connected*

*graphs*with any

*number*

*of*cutpoints.” 98m:05094 05C35 Alon, Noga (IL-TLAV; Tel Aviv); Halperin, Eran (I1L-TLAV-C; Tel Aviv)

*Bipartite*subgraphs ...

##
###
Hamiltonicity in Convex Bipartite Graphs
[article]

2018
*
arXiv
*
pre-print

For a

arXiv:1809.06113v1
fatcat:xsbfav6tpna3no622lq7wfomkm
*connected**graph*,*the*Hamiltonian cycle (path) is a simple cycle (path) that spans all*the*vertices*in**the**graph*. ... We present an infinite*set**of*counterexamples*in*support*of*our claim. ... Acknowledgements:*The*authors wish to appreciate and thank N Narayanan (IIT Madras), C Venkata Praveena (IIITDM Kancheepuram) and P Renjith (IIIT Kottayam) for sharing their insights*on*this problem. ...##
###
Page 6169 of Mathematical Reviews Vol. , Issue 90K
[page]

1990
*
Mathematical Reviews
*

Three inequalities between

*the**domination**number*(*the**maximum**number**of*vertices*of*a*dominating**set*) and*the*chromatic*number**of*a finite undirected*graph*are presented. ...*The*note is concerned with*the*problem*of*finding a partition*of*a given*set**of*vertices A into a minimum possible*number**of**convex**sets*. ...##
###
Restrictions of Minimum Spanner Problems

1997
*
Information and Computation
*

*In*this paper, we completely settle

*the*complexity status

*of*this problem for various values

*of*t,

*on*chordal

*graphs*, split

*graphs*,

*bipartite*

*graphs*and

*convex*

*bipartite*

*graphs*. ... Finally, we provide approximation algorithms for

*the*bandwidth

*minimization*problem

*on*

*convex*

*bipartite*

*graphs*and split

*graphs*using

*the*notion

*of*tree spanners. ] 1997 Academic Press article no. ...

*graphs*, which also appears

*in*[21] ; [22] was subsequently replaced by

*the*current paper with five authors. ...

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