Various Bounds for Liar’s Domination Number

Discussiones Mathematicae Graph Theory

Rad Nader Jafari¹  Mojdeh Doost Ali²  Alimadadi Abdollah³ 

[1] Department of Mathematics Shahrood University of Technology, Shahrood, Iran;Department of Mathematics University of Mazandaran, Babolsar, Iran;Department of Mathematics University of Tafresh, Tafresh, Iran;

关键词: liar’s domination;    diameter;    regular graph;    nordhaus-gaddum;   

DOI  :  10.7151/dmgt.1878

来源: DOAJ

【 摘 要 】

Let G = (V,E) be a graph. A set S ⊆ V is a dominating set if Uv∈S N[v] = V , where N[v] is the closed neighborhood of v. Let L ⊆ V be a dominating set, and let v be a designated vertex in V (an intruder vertex). Each vertex in L ∩ N[v] can report that v is the location of the intruder, but (at most) one x ∈ L ∩ N[v] can report any w ∈ N[x] as the intruder location or x can indicate that there is no intruder in N[x]. A dominating set L is called a liar’s dominating set if every v ∈ V (G) can be correctly identified as an intruder location under these restrictions. The minimum cardinality of a liar’s dominating set is called the liar’s domination number, and is denoted by γLR(G). In this paper, we present sharp bounds for the liar’s domination number in terms of the diameter, the girth and clique covering number of a graph. We present two Nordhaus-Gaddum type relations for γLR(G), and study liar’s dominating set sensitivity versus edge-connectivity. We also present various bounds for the liar’s domination component number, that is, the maximum number of components over all minimum liar’s dominating sets.

【 授权许可】

Unknown