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γ-Excellent, critically dominated, end-dominated, and dot-critical trees are equivalent

Tamara Burton, David P. Sumner
2007 Discrete Mathematics  
A graph is -excellent if every vertex of the graph is contained in some minimum dominating set of the graph. A vertex v is critical in G if the domination number of G − v is smaller than that of G.  ...  In this paper we show that these three properties, along with several others, are equivalent for trees on at least four vertices. We also provide a constructive characterization of these trees.  ...  Clearly, every restrained dominating set for a graph G is also an end-dominating set. The next theorem is from Hattingh and Henning [10] . Theorem 3.1 (Hattingh and Henning [10] ).  ... 
doi:10.1016/j.disc.2006.02.016 fatcat:k7hkttq6pvegremd6wn4nlqpie

Excellent graphs with respect to domination: subgraphs induced by minimum dominating sets

2021 Discrete Mathematics Letters  
A graph G = (V (G), E(G)) is γ-excellent if V (G) is the union of all γ-sets of G, where γ stands for the domination number of G and a γ-set is a dominating set of cardinality γ.  ...  We consider some well-known graphs, including cycles, trees and some cartesian products of two graphs, and for every considered graph we describe its largest set H I for which the graph is H-γ-excellent  ...  The minimum cardinality of a set S which is simultaneously total dominating and restrained dominating in G is called the total restrained domination number γ tr (G) of G.  ... 
doi:10.47443/dml.2020.0052 fatcat:apo7zz5dlbg7xidkhzkloqzgga

Excellent graphs with respect to domination: subgraphs induced by minimum dominating sets [article]

Vladimir Samodivkin
2020 arXiv   pre-print
A graph G=(V,E) is γ-excellent if V is a union of all γ-sets of G, where γ stands for the domination number. Let ℐ be a set of all mutually nonisomorphic graphs and ∅≠ℋ⊊ℐ.  ...  For each of some well known graphs, including cycles, trees and some cartesian products of two graphs, we describe its largest set ℋ⊊ℐ for which the graph is ℋ-γ-excellent.  ...  The minimum cardinality of a set S which is simultaneously total dominating and restrained dominating in G is called the total restrained domination number γ tr (G) of G.  ... 
arXiv:2010.03219v1 fatcat:qdopeiq6gveybf2b3nkjed2pfy

On weakly connected domination in graphs II

Gayla S. Domke, Johannes H. Hattingh, Lisa R. Markus
2005 Discrete Mathematics  
Lastly, we constructively characterize the trees T in which every vertex belongs to some weakly connected dominating set of cardinality wc (T ).  ...  We provide a constructive characterization of trees T for which (T ) = wc (T ).  ...  Characterizations of ( , )-trees include ( , i)-trees [1, 6] , trees with equal independent domination and restrained domination numbers, trees with equal independent domination and weak domination numbers  ... 
doi:10.1016/j.disc.2005.10.006 fatcat:ozdpz7nugzc7th4tmvtkjue4ju

Page 137 of Photo-Era Magazine : The American Journal of Photography Vol. 37, Issue 3 [page]

1916 Photo-Era Magazine : The American Journal of Photography  
“Winter Shadows” is excellent in its blue-black tone, but it seems to lack a center of interest. H. J.  ...  “HIS COMPANIONS, INNOCENCE AND HEALTH, AND HIS BEST RICHES, IGNORANCE OF WEALTH > THIRD PRIZE — BEGINNERS CONTEST “The Cloud Over the Sun” is well named; it cer- tainly dominates the picture.  ... 

Page 921 of American Anthropologist Vol. 76, Issue 4 [page]

1974 American Anthropologist  
A Green Tree and a Dry Tree. CARTER WILSON. New York: Macmillan, 1972. 300 pp., maps, notes. $7.95 (cloth). Reviewed by GARY H.  ...  This observation in no way detracts from the re- viewer’s opinion as to the general excellence of this work.  ... 

Page 146 of American Journal of Theology Vol. 14, Issue 1 [page]

1910 American Journal of Theology  
Spence’s proposals are so elaborate as to hinder their excellent purpose.  ...  In the rise of this principle as a dominating factor in our private life is the assurance of eventual reunion. ‘It must be evident to all,” says Dr.  ... 

Upper bounds for domination related parameters in graphs on surfaces

Vladimir Samodivkin
2016 AKCE International Journal of Graphs and Combinatorics  
In this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic  ...  We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum  ...  The restrained domination number of G, denoted by γ r (G), is the minimum cardinality of a restrained dominating set of G.  ... 
doi:10.1016/j.akcej.2016.06.007 fatcat:i7xnfbn2pbhozf6iltjbzi5try

Page 439 of Southern Review Vol. 5, Issue 10 [page]

1830 Southern Review  
The dominant party deter- mined to discipline his mind by confining his body.  ...  For now that God hath cut this tree of Para- dise down from its seat of earth, yet so the dead trunk may support x part of the declining temple, or, at least, serve to kindle the fire on the altar.  ... 

Social hierarchy under different criteria in groups of squirrel monkeys,Saimiri sciureus

Fernando Alvarez
1975 Primates  
The ranking of the subjects was much the same for Restraining and Genital display.  ...  Restraining and Genital display showed quasi-linear hierarchies, with males on top and females at the bottom of them.  ...  In addition to the varied ways in which dominance could be achieved, it was also pointed out that the social functions of highly dominant animals were far richer than is implied by defining dominance in  ... 
doi:10.1007/bf02382741 fatcat:2yv5mwvbrrdipejdspr65r3d6e

Page 159 of Photo-Era Magazine : The American Journal of Photography Vol. 41, Issue 3 [page]

1918 Photo-Era Magazine : The American Journal of Photography  
“The Cloud Over the Sun” is well named; it cer- tainly dominates the picture.  ...  Even then there would remain an excellent cloud- effect, occupying fully two-thirds of the picture-area, and the spacing of the whole composition would be improved.  ... 

The Effect of Fertilizers on Biomass and Biodiversity on a Semi-Arid Grassland of Northern China

Zongyong Tong, Guoling Quan, Liqiang Wan, Feng He, Xianglin Li
2019 Sustainability  
Leymus chinensis was first divided into its own group and then separated into four groups via cluster tree analysis.  ...  All data were processed, and the graphs were plotted by Microsoft Excel 2010.  ...  Figure 3 . 3 Species cluster tree map of importance values. Figure 3 . 3 Species cluster tree map of importance values.  ... 
doi:10.3390/su11102854 fatcat:hzerro7psncrhadxbcimqd3aby

On independent [1,2]-sets in trees

Sahar A. Aleid, José Cáceres, Maria Luz Puertas
2018 Discussiones Mathematicae Graph Theory  
We follow the concept of excellent tree and characterize the family of trees such that any vertex belongs to some independent [1, 2]-set.  ...  An [1, k]-set S in a graph G is a dominating set such that every vertex not in S has at most k neighbors in it.  ...  This concept has been studied in the family of trees for different domination-type properties such as domination, irredundance and independence [6, 9] , restrained domination [7] and total domination  ... 
doi:10.7151/dmgt.2029 fatcat:j2mumtbqmjghhppslemzztfxbu

On independent [1,2]-sets in trees [article]

Sahar Aleid, Jose Caceres, Maria Luz Puertas
2015 arXiv   pre-print
We follow the concept of excellent tree and characterize the family of trees such that any vertex belong to some independent [1,2]-set.  ...  An independent [1,k]-set S in a graph G is a dominating set which is independent and such that every vertex not in S has at most k neighbors in it.  ...  This concept has been studied in the family of trees for different domination-type properties such as domination, irredundance and independence [4, 7] , restrained domination [5] and total domination  ... 
arXiv:1511.09262v1 fatcat:x5x6sxcfbbgjdhf4oqsquzbsgm

Page 704 of Eclectic Magazine of Foreign Literature Vol. 45, Issue 5 [page]

1887 Eclectic Magazine of Foreign Literature  
But if Rossetti has not magically in¬ terpreted external nature, if the literary instinct in him occasionally too marked¬ ly dominates the purely poetic, there is one point wherein he excels any contem  ...  urgent wood-notes soar, to turn to the unlabored and impulsive strain of the Scottish singer— The mavis sings fu blithely On ilka leafy bough ; or to the English poet’s— Hark \ where my blossomed pear-tree  ... 
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