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On a Nordhaus-Gaddum type problem for independent domination

1995
*
Discrete Mathematics
*

Let i(G) (i(G), respectively) be the

doi:10.1016/0012-365x(94)00202-t
fatcat:g555gwmecbeg7j5pvddam57d2q
*independent**domination*number (i.e. smallest cardinality of*a*maximal*independent*vertex subset) of the p-vertex graph G (the complement G of G, respectively). ... The research was partially supported by the Canadian Natural Sciences and Engineering Research Council and by the South African Foundation*for*Research Development. ... The calculations of extremum values of q(G) + q((7) and q(G)q((7) taken over all p-vertex graphs G are known as*Nordhaus*-*Gaddum**type**problems*, due to the results of [3] , in which*Nordhaus*and*Gaddum*...##
###
Page 4775 of Mathematical Reviews Vol. , Issue 98H
[page]

1998
*
Mathematical Reviews
*

Summary: “

*A**Nordhaus*-*Gaddum*-*type*result is*a*(tight) lower or upper bound*on*the sum or product of*a*parameter of*a*graph and its complement. ... {*For*the entire collection see MR 98f:05001.} 98h:05106 05C35 Sivagurunathan, S. (6-I1TM; Chennai Some*Nordhaus*-*Gaddum*-*type*results. ...##
###
Page 2091 of Mathematical Reviews Vol. , Issue 98D
[page]

1998
*
Mathematical Reviews
*

“

*Nordhaus*-*Gaddum*results” generalize those in an earlier paper by E.*A*.*Nordhaus*and J. W.*Gaddum*[Amer. Math. ... In this paper, we obtain some bounds*for*»/(G).*Nordhaus*-*Gaddum*-*type*results are also found.” 98d:05077 05C35 0SC1S 05C70 Sivagurunathan, S. (6-IITK; Kanpur); Mohanty, S. P. ...##
###
A note on Nordhaus–Gaddum inequalities for domination

2004
*
Discrete Applied Mathematics
*

Let u(G) be

doi:10.1016/s0166-218x(03)00200-2
fatcat:mos6kyf3x5gtpjjarl3fx3gdyy
*a*graph parameter, the calculations of extremum values of u(G) + u( G) and u(G)u( G) taken over all n-vertex graphs G are known as*Nordhaus*-*Gaddum**type**problems*, due to the results of [5] ... Let W 1 be*a*maximum sized*independent*set*for*W in G, then |W 1 | ¿ ( G) − 2 ¿ k. Otherwise ( G) 6 |W 1 | + 2 ¡ k + 2,*a*contradiction. Now W 1 ; W 1 ∪ {x}; W 1 ∪ {y} are*independent*sets in G. ...##
###
Nordhaus-Gaddum type inequalities for multiple domination and packing parameters in graphs
[article]

2017
*
arXiv
*
pre-print

We study the

arXiv:1610.06419v2
fatcat:ci3vtv3y4fgsxhkd3b63gm4h54
*Nordhaus*-*Gaddum**type*results*for*(k-1,k,j) and k-*domination*numbers of*a*graph G and investigate these bounds*for*the k-limited packing and k-total limited packing numbers in graphs. ... As the special case (k-1,k,j)=(1,2,0) we give an upper bound*on*dd(G)+dd(G) stronger than that presented by Harary and Haynes (1996). ... Moreover, by the symmetry we have The first*Nordhaus*-*Gaddum**type*inequality*for*the sum of the total*domination*numbers of*a*graph and its complement was given in [2] . Henning et al. ...##
###
Page 784 of Mathematical Reviews Vol. , Issue 97B
[page]

1997
*
Mathematical Reviews
*

First we give

*a*brief survey of*Nordhaus*-*Gaddum*results*for*several*domination*-related parameters. Then we present new inequalities of this*type*involving double domi- nation. ...*A*direct result of our bounds*for*double*domination*in complementary graphs is*a*new*Nordhaus*-*Gaddum*inequality*for*open*domination*improving known bounds*for*the case when both G and its complement have ...##
###
On three outer-independent domination related parameters in graphs
[article]

2018
*
arXiv
*
pre-print

We prove some

arXiv:1812.10946v1
fatcat:yjcrn2g5dbbzbm6ndfdora42tu
*Nordhaus*-*Gaddum**type*inequalities, derive their computational complexity and present several bounds*for*them. ... Let G be*a*graph and let S⊆ V(G). The set S is*a*double outer-*independent**dominating*set of G if |N[v]∩ D|≥2,*for*all v∈ V(G), and V(G)∖ S is*independent*. ... We begin with the*Nordhaus*-*Gaddum**type*inequalities*for*the TOID number of graphs. ...##
###
Note on Nordhaus-Gaddum Problems for Colin de Verdière type Parameters

2013
*
Electronic Journal of Combinatorics
*

(G)+\xi(\overline{G})\le b_\xi | G|$

doi:10.37236/2570
fatcat:otqx5a4yjzaddmljuosiecz5rq
*for*all graphs $G$, and $\nu$ and $\xi$ are Colin de Verdiere*type*graph parameters. ... The*Nordhaus*-*Gaddum*sum lower bound*for*$\nu$ and $\xi$ is conjectured to be $|G| - 2$, and if these parameters are replaced by the maximum nullity $M(G)$, this bound is called the Graph Complement Conjecture ... Introduction*Nordhaus*-*Gaddum**problems*have been studied*for*many different graph parameters, including chromatic number,*independence*number,*domination*number, Hadwiger number, etc. ...##
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The inverse strong non-split r-domination number of a graph

2010
*
International Journal of Engineering, Science and Technology
*

*Nordhaus*-

*Gaddum*

*type*results are also obtained

*for*this new parameter. ... We characterize graphs

*for*which γ sns r(G) + γ′ sns r(G) = n, where γ sns r(G) is the strong non-split r-

*domination*number of G. We get many bounds

*on*γ′ sns r(G). ...

*Nordhaus*-

*Gaddum*

*type*results (

*Nordhaus*and

*Gaddum*, 1956) 4.1.1 Theorem Let G and G be connected complementary graphs with δ r (G) = 1. ...

##
###
Packing parameters in graphs: New bounds and a solution to an open problem
[article]

2017
*
arXiv
*
pre-print

By applying the Mantel's theorem, We give upper bounds

arXiv:1705.08667v3
fatcat:m6eaegna65gtten5zygdwrwdn4
*on*packing and open packing numbers of triangle-free graphs along with characterizing the graphs*for*which the equalities hold and exhibit sharp*Nordhaus*-*Gaddum*...*type*inequalities*for*packing numbers. ... Diameter and*Nordhaus*-*Gaddum*inequalities*for*packing number Many results in*domination*theory have relationship with the diameter of graphs (see [4] ). ...##
###
On the Total Outer k-Independent Domination Number of Graphs

2020
*
Mathematics
*

In addition, we give several

doi:10.3390/math8020194
fatcat:rpplta7sjjfqpj7qxagagqxp4e
*Nordhaus*–*Gaddum**type*results. Finally, we prove that computing the total outer k-*independent**domination*number of*a*graph G is an NP-hard*problem*. ...*For*instance, we give several closed relationships between this novel parameter and other*ones*related to*domination*and*independence*in graphs. ... Acknowledgments: We would like to thank the referees*for*their careful reading of the manuscript and several useful comments that have helped us to improve the presentation of the paper. ...##
###
A Study on Connected Domination Transition Number of a Graph

2018
*
International Journal of Engineering & Technology
*

We also attempted to derive few

doi:10.14419/ijet.v7i4.10.20916
fatcat:7bgxutft2vgkhpnogemd562sma
*Nordhaus*–*Gaddum*(NG)*type*results*for*the same. ... Recently, some researchers attempted*on*transition of the*domination*parameter into*a*connected*one*. ... Among the other*types*of inequalities available in literature, the*Nordhaus*-*Gaddum*(NG)*type*inequalities are considered to be peculiar since they view*a*single parameter in two different ways*one*with ...##
###
Page 5762 of Mathematical Reviews Vol. , Issue 95j
[page]

1995
*
Mathematical Reviews
*

Mynhardt,

*On**a**Nordhaus*-*Gaddum**type**problem**for**independent**domination*(199-205). ... of the same length (347-353); Jian Fang Wang and Ting Zai Tang,*On*arithmetic numbering of cy- cles (355-362); Zhi Jian Wang [Zhi Jian Wang'], Some results*on*arboricity parameters of the*Nordhaus*-*Gaddum*...##
###
Page 5981 of Mathematical Reviews Vol. , Issue 2004h
[page]

2004
*
Mathematical Reviews
*

We exploit this equality to show that computation of con is NP-complete, provide

*a**Nordhaus*-*Gaddum**type*bound*for*con and produce*a*Ramsey*type*theorem*for*con. ... Extensions to other parameters*a*,(G) are also derived.” 2004h:05089 —05C69 Cockayne, E. J. (3-VCTR-MS; Victoria, BC)*Nordhaus*-*Gaddum*results*for*open irredundance. (English summary) J. Combin. Math. ...##
###
Double Roman and double Italian domination

2021
*
Discussiones Mathematicae Graph Theory
*

Let G be

doi:10.7151/dmgt.2399
fatcat:4znkesc2affwnhzk2cew5c7r6a
*a*graph with vertex set ...*Nordhaus*-*Gaddum**Type*Results Results of*Nordhaus*-*Gaddum**type*study the extreme values of the sum or product of*a*parameter*on**a*graph and its complement. ... Jafari Rad and Rahbani [13] presented*Nordhaus*-*Gaddum**type*inequalities*for*the double Roman*domination*number. ...
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