Abstract
We solve several conjectures and open problems from a recent paper by Acharya [2]. Some of our results are relatives of the Nordhaus-Gaddum theorem, concerning the sum of domination parameters in hypergraphs and their complements. (A dominating set in H is a vertex set D X such that, for every vertex x? X\D there exists an edge E ? E with x ? E and E?D ??.) As an example, it is shown that the tight bound ??(H)+??(H) ? n+2 holds in hypergraphs H = (X, E) of order n ? 6, where H is defined as H = (X, E) with E = {X\E | E ? E}, and ?? is the minimum total cardinality of two disjoint dominating sets. We also present some simple constructions of balanced hypergraphs, disproving conjectures of the aforementioned paper concerning strongly independent sets. (Hypergraph H is balanced if every odd cycle in H has an edge containing three vertices of the cycle; and a set S X is strongly independent if |S?E|? 1 for all E ? E.).
Highlights
In graphs, the theory of dominating sets is extensively studied, with well over 1000 publications, see e.g. the book [6] and the recent papers [3, 7]
We shall prove that γ(H)+γ−1(H)+γ(H)+γ−1(H) ≤ max{8, n+2} holds, which is tight for all n ≥ 4 for γγ(H) + γγ(H). (It follows by definition that γγ(H) ≤ γ(H) + γ−1(H), cf. [2].) min{γγ(H), γγ(H)} = min{γ(H) + γ−1(H), γ(H) + γ−1(H)} ≤ 4
We disprove Conjecture 3.8 of [2] (which stated that γ(H) = γi(H) implies γγ(H) = γ(H) + γ−1(H) for every connected H) by giving an infinite family of counterexamples
Summary
The theory of dominating sets is extensively studied, with well over 1000 publications, see e.g. the book [6] and the recent papers [3, 7]. It is equivalent to assume that a set does not contain any edge, or any two of its vertices are non-adjacent. In hypergraphs, the former condition is weaker than the latter. A hypergraph is said to be balanced if every odd cycle in H has an edge containing three vertices of the cycle. We observe that Problem 2 of [2], about the characterization of hypergraphs having two disjoint maximal strongly independent sets, is reducible to the same problem on graphs
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