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Bounds on the strong domination number

Dieter Rautenbach
2000 Discrete Mathematics  
Whereas for the strong domination number there are many structural restrictions on the graph (see [2] and [5] ) which affect this parameter, similar assumptions have no effect on the weak domination  ...  Preliminary Results In [6] the authors also introduced a strong domination number rst( G) which differs from 'Yw( G) only by the opposite degree demand.  ... 
doi:10.1016/s0012-365x(99)00248-4 fatcat:rbnngfd7mjfp7gyphrrhaldjma

THE STRONG (WEAK) VV-DOMINATING SET OF A GRAPH

Sayinath Udupa, R. S. Bhat
2019 Malaysian Journal of Science  
The strong vv-domination number γ ( ) is the order of the minimum strong vv-dominating set of G. Similarly weak vvdomination number γ ( ) is defined.  ...  A set of vertices is said to be strong vv-dominating set if each vertex outside the set is strongly vv-dominated by at least one vertex inside the set.  ...  Several bounds for strong block domination parameters are obtained.  ... 
doi:10.22452/mjs.vol38no3.3 fatcat:jhi5p4bt2zfjdeuu22wctgyeki

Bounds on weak and strong total domination in graphs

M.H. Akhbari, N. Jafari Rad
2016 Electronic Journal of Graph Theory and Applications  
The strong total domination number γ st (G) of G is the minimum cardinality of a strong total dominating set of G. We present some bounds on weak and strong total domination number of a graph.  ...  The weak total domination number γ wt (G) of G is the minimum cardinality of a weak total dominating set of G.  ...  Similarly we obtain the following. Sharp bounds on the strong total domination number in trees Chellali et al.  ... 
doi:10.5614/ejgta.2016.4.1.10 fatcat:3gj4fctlkzdflerdatres33m4y

On the global offensive alliance number of a graph

J.M. Sigarreta, J.A. Rodríguez
2009 Discrete Applied Mathematics  
In this paper we obtain several tight bounds on γ o (Γ ) and γ co (Γ ) in terms of several parameters of Γ . The case of strong alliances is studied by analogy.  ...  In the case of strong offensive alliance, strict majority is required. An alliance S is called global if it affects every vertex in V \ S, that is, S is a dominating set of Γ .  ...  Notice that the bound γ (Γ )+n 2 on γ o (Γ ) is never worse than the bound n+α(Γ ) 2 . The advantage of the second one is when there is known the independence number but not the domination number.  ... 
doi:10.1016/j.dam.2008.02.007 fatcat:nxb3gzs4h5gu5ha6ksldnm4gei

On the global offensive alliance number of a graph [article]

J. M. Sigarreta, J. A. Rodriguez
2006 arXiv   pre-print
The offensive alliance number a_o(Γ) (respectively, strong offensive alliance number a_ô(Γ)) is the minimum cardinality of an offensive (respectively, strong offensive) alliance in Γ.  ...  In this paper we obtain several tight bounds on γ_o(Γ) and γ_ô(Γ) in terms of several parameters of Γ.  ...  The study of offensive alliances was initiated by Favaron et al. in [2] where were derived several bounds on the offensive alliance number and the strong offensive alliance number.  ... 
arXiv:math/0602432v1 fatcat:npkvr57tmrfkzlxxyxq2zv2ecq

Strong Transversals in Hypergraphs and Double Total Domination in Graphs

Michael A. Henning, Anders Yeo
2010 SIAM Journal on Discrete Mathematics  
The set T is a strong transversal in H if T contains at least two vertices from every edge of H. The strong transversal number τ s (H) of H is the minimum size of a strong transversal in H.  ...  The minimum cardinality of a double total dominating set of G is the double total domination number γ ×2,t (G) of G. Let G be a connected graph of order n with minimum degree at least three.  ...  The set T is a strong transversal in H if T contains at least two vertices from every edge of H. The strong transversal number τ s (H) of H is the minimum size of a strong transversal in H.  ... 
doi:10.1137/090777001 fatcat:7icwcscn5zeljaxrdnxzyaiona

Probabilistic Bounds On Weak And Strong Total Domination In Graphs

M.H.Akhbari
2016 Zenodo  
A set D of vertices in a graph G = (V,E) is a total dominating set if every vertex of G is adjacent to some vertex in D.  ...  We obtain probabilistic upper bounds on weak and strong total domination number of a graph. §2. Results We adopt the notations of [1] .  ...  The strong total domination number γ st (G) of G is the minimum cardinality of a strong total dominating set of G.  ... 
doi:10.5281/zenodo.815745 fatcat:bblmcspxcjenflzayfffwb4frm

On fuzzy dominator coloring in fuzzy graphs

R. Jahir Hussain, K. S. Kanzul Fathima
2015 Applied Mathematical Sciences  
It is proved that fuzzy dominator chromatic number equals fuzzy chromatic number if at least one node of G has strong degree n-1.  ...  Bounds for fuzzy dominator chromatic number and necessary and sufficient condition for fuzzy dominator chromatic number to be 2 and n has been found.  ...  Then adding the edge u1un will preserve the FDC of Pn in Cn. Bounds of Fuzzy Dominator Chromatic number Let G be a connected fuzzy graph where the number of nodes is n2.  ... 
doi:10.12988/ams.2015.4121015 fatcat:ddgihlowj5fstdqpeeaqudeofy

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

B.K Ameenal, R Selvakumar
2010 International Journal of Engineering, Science and Technology  
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).  ...  In this paper, we define the notions of inverse strong non-split r-dominating set and inverse strong non-split r-domination number γ′ sns r(G) of a graph G.  ...  Then r-clique is denoted by r ω (G) and is defined by the maximum order of r-complete vertices. Bounds on γ sns ′r(G) Here, we get bounds on γ sns ′r(G) through the following theorems.  ... 
doi:10.4314/ijest.v2i1.59102 fatcat:fq7562dafbd2df52j5r6sif4tq

Strong Geodetic Domination of Graphs

2019 VOLUME-8 ISSUE-10, AUGUST 2019, REGULAR ISSUE  
Here the domination concept is combined with strong geodetic concept resulting in Strong geodetic domination of graphs and few results are derived.  ...  The problem to find a where all points of the graph are covered by a unique fixed geodesic between the pair of points in is called the strong geodetic problem.  ...  For , where the GD number is 2, while the SGD number is equal to -. Thus, the SGD number is not in the near bound of the GD number.  ... 
doi:10.35940/ijitee.l2503.1081219 fatcat:rdy3jjv72remhlgpjgjybtxixa

On strong (weak) independent sets and vertex coverings of a graph

S.S. Kamath, R.S. Bhat
2007 Discrete Mathematics  
The strong (weak) vertex covering number s = s (G) (w = w (G)) is the minimum cardinality of an SVC (WVC). In this paper, we investigate some relationships among these four new parameters.  ...  Further, we show that s p − and w p − and find a necessary and sufficient condition to attain the upper bound, characterizing the graphs which attain these bounds.  ...  Acknowledgement The authors are grateful to the referees for their valuable comments and suggestions.  ... 
doi:10.1016/j.disc.2006.07.040 fatcat:e7wzrbxha5crnjt3opbdldq47e

Page 65 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews  
The strong domination number 7,,(G) is defined as the minimum cardinality of a strong dominating set and was introduced by E. Sampathkumar and L.  ...  Gaz. 49 (1965), 290-293; MR 32 #4025] that every n-vertex tournament is dominated by at most lgn —Iglgn +2 vertices.” 2001a:05114 05C69 Rautenbach, Dieter (D-AACH-M2; Aachen) Bounds on the strong domination  ... 

BONDAGE AND NON-BONDAGE NUMBER OF A FUZZY GRAPH

A.N. Gani, K.P. Devi, M. Akram
2015 International Journal of Pure and Applied Mathematics  
The bondage number b(G) and non-bondage number b n (G) of a fuzzy graph G are defined. The upper bound for both b(G) and b n (G) are given. Also some results on b(G) and b n (G) are discussed.  ...  The exact values of b(G) and b n (G) are determined for several classes of fuzzy graphs.  ...  In G, all arcs are strong arcs. Thus the total number of (strong) arcs in G are n(n − 1)/2. We know that γ(G) = 1. Each node will dominate all other nodes.  ... 
doi:10.12732/ijpam.v103i2.7 fatcat:ts2c6gwc7ngknh5tdhwth2fnvi

Offensive alliances in cubic graphs

J. A. Rodriguez, J. M. Sigarreta
2006 International Mathematical Forum  
In the case of strong offensive alliance, strict majority is required. An alliance S is called Mathematics Subject Classification: 05C69; 15C05  ...  An offensive alliance in a graph Γ = (V, E) is a set of vertices S ⊂ V where for every vertex v in its boundary it holds that the majority of vertices in v's closed neighborhood are in S.  ...  The particular case of global (strong) defensive alliance was investigated in [5] where several bounds on the global (strong) defensive alliance number were obtained.  ... 
doi:10.12988/imf.2006.06152 fatcat:6i4h3gvaxzhsnm6alhro76ydfi

Inverse Independence Number of a Graph

P. G.Bhat, Surekha R Bhat
2012 International Journal of Computer Applications  
The inverse independence Number β 0 -1 (G) = max { S : S is an inverse independent set of G}.We find few bounds on inverse domination number and also initiate the study of the inverse independence number  ...  giving few bounds on inverse independence number of a graph.  ...  BOUNDS ON INVERSE DOMINATION NUMBER The concept of inverse domination is introduced by Kulli V.R and Sigarakanti S.C [9] . Let D be a  -set of G.  ... 
doi:10.5120/5687-7734 fatcat:ynifewxh2fd3nh4u7ly5ldqe6e
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