An upper bound on the 2-outer-independent domination number of a tree

2011 ◽  
Vol 349 (21-22) ◽  
pp. 1123-1125 ◽  
Author(s):  
Marcin Krzywkowski
Keyword(s):  
Upper Bound ◽  
Domination Number ◽  
Independent Domination

scholarly journals Independent [1,2]-number versus independent domination number

2017 ◽  
Vol 25 (3) ◽  
pp. 5-24
Author(s):  
Sahar A. Aleid ◽  
Mercè Mora ◽  
María Luz Puertas
Keyword(s):  
Upper Bound ◽  
Domination Number ◽  
Dominating Sets ◽  
Local Conditions ◽  
Additional Requirement ◽  
Graph Class ◽  
Independent Domination ◽  
The Relationship

Abstract A [1; 2]-set S in a graph G is a vertex subset such that every vertex not in S has at least one and at most two neighbors in it. If the additional requirement that the set be independent is added, the existence of such sets is not guaranteed in every graph. In this paper we provide local conditions, depending on the degree of vertices, for the existence of independent [1; 2]-sets in caterpillars. We also study the relationship between independent [1; 2]-sets and independent dominating sets in this graph class, that allows us to obtain an upper bound for the associated parameter, the independent [1; 2]-number, in terms of the independent domination number.

An upper bound for the double outer-independent domination number of a tree

2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Marcin Krzywkowski
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Domination ◽  
Independent Dominating Set

AbstractA vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph

scholarly journals Independence and domination on shogiboard graphs

2017 ◽  
Vol 4 (8) ◽  
pp. 25-37 ◽  
Author(s):  
Doug Chatham
Keyword(s):  
Independence Number ◽  
Domination Number ◽  
The Other ◽  
Independent Domination

Abstract Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n×n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, which moves like a bishop and rook from chess. We show that the independence number for the dragon kings graph equals the independence number for the queens graph. We show that the (independent) domination number of the dragon kings graph is n − 2 for 4 ≤ n ≤ 6 and n − 3 for n ≥ 7. For the dragon horses graph, we show that the independence number is 2n − 3 for n ≥ 5, the domination number is at most n−1 for n ≥ 4, and the independent domination number is at most n for n ≥ 5.

scholarly journals An upper bound on the domination number of n-vertex connected cubic graphs

2009 ◽  
Vol 309 (5) ◽  
pp. 1142-1162 ◽  
Author(s):  
A.V. Kostochka ◽  
B.Y. Stodolsky
Keyword(s):  
Upper Bound ◽  
Domination Number ◽  
Cubic Graphs

scholarly journals Hop Domination in Graphs-II

2015 ◽  
Vol 23 (2) ◽  
pp. 187-199
Author(s):  
C. Natarajan ◽  
S.K. Ayyaswamy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
The Family ◽  
Independent Domination ◽  
Domination In Graphs ◽  
Domination Numbers

Abstract Let G = (V;E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V - S, there exists u ∈ S such that d(u; v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

A note on improved upper bounds on the transversal number of hypergraphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950004
Author(s):  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Size ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

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