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On a Nordhaus-Gaddum type problem for independent domination

E.J. Cockayne, G. Fricke, C.M. Mynhardt
1995 Discrete Mathematics  
Let i(G) (i(G), respectively) be the 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  ... 
doi:10.1016/0012-365x(94)00202-t fatcat:g555gwmecbeg7j5pvddam57d2q

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

Shan Erfang, Dang Chuangyin, Kang Liying
2004 Discrete Applied Mathematics  
Let u(G) be 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.  ... 
doi:10.1016/s0166-218x(03)00200-2 fatcat:mos6kyf3x5gtpjjarl3fx3gdyy

Nordhaus-Gaddum type inequalities for multiple domination and packing parameters in graphs [article]

D.A. Mojdeh, Babak Samadi, Lutz Volkmann
2017 arXiv   pre-print
We study the 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.  ... 
arXiv:1610.06419v2 fatcat:ci3vtv3y4fgsxhkd3b63gm4h54

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]

Doost Ali Mojdeh, Iztok Peterin, Babak Samadi, Ismael G. Yero
2018 arXiv   pre-print
We prove some 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.  ... 
arXiv:1812.10946v1 fatcat:yjcrn2g5dbbzbm6ndfdora42tu

Note on Nordhaus-Gaddum Problems for Colin de Verdière type Parameters

Wayne Barrett, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben
2013 Electronic Journal of Combinatorics  
(G)+\xi(\overline{G})\le b_\xi | G|$ 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.  ... 
doi:10.37236/2570 fatcat:otqx5a4yjzaddmljuosiecz5rq

The inverse strong non-split r-domination number of a graph

B.K Ameenal, R Selvakumar
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.  ... 
doi:10.4314/ijest.v2i1.59102 fatcat:fq7562dafbd2df52j5r6sif4tq

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

D.A. Mojdeh, Babak Samadi
2017 arXiv   pre-print
By applying the Mantel's theorem, We give upper bounds 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] ).  ... 
arXiv:1705.08667v3 fatcat:m6eaegna65gtten5zygdwrwdn4

On the Total Outer k-Independent Domination Number of Graphs

Abel Cabrera Martínez, J. C. Hernández-Gómez, E. Parra Inza, José M. Sigarreta
2020 Mathematics  
In addition, we give several NordhausGaddum 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.  ... 
doi:10.3390/math8020194 fatcat:rpplta7sjjfqpj7qxagagqxp4e

A Study on Connected Domination Transition Number of a Graph

Kaspar S, B. Gayathri, D. Kalpanapriya, M. P. Kulandaivel3
2018 International Journal of Engineering & Technology  
We also attempted to derive few NordhausGaddum (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  ... 
doi:10.14419/ijet.v7i4.10.20916 fatcat:7bgxutft2vgkhpnogemd562sma

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

Lutz Volkmann
2021 Discussiones Mathematicae Graph Theory  
Let G be 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.  ... 
doi:10.7151/dmgt.2399 fatcat:4znkesc2affwnhzk2cew5c7r6a
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