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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...
In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turán [8], we present a sharp lower bound on Kr+1-free graphs for r ≥ 2. Applying the concept of total limited packing we bound the signed total domination number of G with δ(G) ≥ 3 from above by [...] . Also, we prove that γst(T) ≤ n − 2(s − s′) for any tree T of order n, with s support vertices and s′ support vertices...
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