STRONG AND WEAK INDEPENDENT DOMINATION NUMBERS OF HALIN GRAPH

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.

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Domination critical graphs with higher independent domination numbers

Journal of Graph Theory ◽

10.1002/(sici)1097-0118(199605)22:1<9::aid-jgt2>3.0.co;2-s ◽

1996 ◽

Vol 22 (1) ◽

pp. 9-14 ◽

Cited By ~ 7

Author(s):

S. Ao ◽

E.J. Cockayne ◽

G. MacGillivray ◽

C.M. Mynhardt

Keyword(s):

Critical Graphs ◽

Independent Domination ◽

Domination Numbers

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On trees with equal 2-domination and 2-outer-independent domination numbers

Indian Journal of Pure and Applied Mathematics ◽

10.1007/s13226-015-0126-7 ◽

2015 ◽

Vol 46 (2) ◽

pp. 191-195 ◽

Cited By ~ 1

Author(s):

Marcin Krzywkowski

Keyword(s):

Independent Domination ◽

Domination Numbers

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Lower bounds on the Roman and independent Roman domination numbers

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm151112023c ◽

2016 ◽

Vol 10 (1) ◽

pp. 65-72 ◽

Cited By ~ 7

Author(s):

Mustapha Chellali ◽

Teresa Haynes ◽

Stephen Hedetniemi

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Minimum Weight ◽

Domination Number ◽

Roman Domination ◽

Roman Domination Number ◽

Roman Dominating Function ◽

Independent Domination ◽

Dominating Function ◽

Domination Numbers

A Roman dominating function (RDF) on a graph G is a function f : V (G) ? {0,1,2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of a Roman dominating function is the sum f(V) = ?v?V f(v), and the minimum weight of a Roman dominating function f is the Roman domination number ?R(G). An RDF f is called an independent Roman dominating function (IRDF) if the set of vertices assigned positive values under f is independent. The independent Roman domination number iR(G) is the minimum weight of an IRDF on G. We show that for every nontrivial connected graph G with maximum degree ?, ?R(G)? ?+1/??(G) and iR(G) ? i(G) + ?(G)/?, where ?(G) and i(G) are, respectively, the domination and independent domination numbers of G. Moreover, we characterize the connected graphs attaining each lower bound. We give an additional lower bound for ?R(G) and compare our two new bounds on ?R(G) with some known lower bounds.

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