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Title of Journal: J Comb Optim

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Abbravation: Journal of Combinatorial Optimization

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Springer US

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DOI

10.1002/2014gl060470

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1573-2886

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Disjunctive total domination in graphs

Authors: Michael A Henning Viroshan Naicker
Publish Date: 2014/11/15
Volume: 31, Issue: 3, Pages: 1090-1110
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Abstract

Let G be a graph with no isolated vertex In this paper we study a parameter that is a relaxation of arguably the most important domination parameter namely the total domination number gamma tG A set S of vertices in G is a disjunctive total dominating set of G if every vertex is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it The disjunctive total domination number gamma d tG is the minimum cardinality of such a set We observe that gamma d tG le gamma tG We prove that if G is a connected graph of order n ge 8 then gamma d tG le 2n1/3 and we characterize the extremal graphs It is known that if G is a connected clawfree graph of order n then gamma tG le 2n/3 and this upper bound is tight for arbitrarily large n We show this upper bound can be improved significantly for the disjunctive total domination number We show that if G is a connected clawfree graph of order n 14 then gamma d tG le 4n/7 and we characterize the graphs achieving equality in this boundResearch of the First Author supported in part by the South African National Research Foundation and the University of Johannesburg Research of the Second Author supported in part by the University of Johannesburg Rhodes University and the ClaudeLeon Foundation

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