eu EQUIVALENCE OF TWO WIDELY RESEARCHED PROBLEMS IN HYPERGRAPH THEORY

The problems of enumerating (i) all the minimal transversals and (ii) all the minimal dominating sets, in a given hypergraph, have received a lot of attention because of their applications in Computer Science. This article explores the possibilities of these two problems being solution-wise equivalent that is, each solution to one of them being a solution to the other in the domain of Sperner hypergraphs, culminating in identifying the only class of such hypergraphs in which the equivalence holds. AMS Subject Classification: 05C65


Introduction
The cardinality (or, size) [8] of a finite set V is denoted by | V |.The power set of V is the set of all subsets (including the empty set φ) of V , and is denoted by 2 V .The set of all nonempty subsets of V is denoted by 2 V * ; that is, 2 V * = 2 V − {φ}.
A simple hypergraph [2] is an ordered couple H = (V, E) where: (i) V is a nonempty finite set and (ii) E is a set of nonempty subsets of V such that X∈E X = V .Each member of V is a vertex in H; and each member of E is a hyperedge (or, an edge) in H.A hyperedge X with | X |= 1 is a loop.H is loop-free if | X |> 1 for every hyperedge X.Two distinct vertices x and y are adjacent if there is a hyperedge that contains both x and y.H is Sperner if no hyperedge is a subset of another.Sperner hypergraphs are necessarily simple, though not conversely [3].
The problem of identifying the minimal transversals in a given hypergraph H will be denoted by HY P − T RAN S − H, and the problem of identifying the minimal dominating sets in H will be denoted by HY P − DOM − H.
The hypergraphs considered in this article are all assumed loop-free and of the Sperner type.A given hypergraph H will be assumed to be the ordered couple H = (V, E) unless alternate notations are explicit.This research is of theoretical interest, motivations for it coming from: (i) minimal transversals and related problems, dealt with in [4]; (ii) minimal dominating sets and related topics, covered in [6], and (iii) transversals and dominating sets, treated in [5].
The following polynomial-time equivalence is considered in [6]: If P and Q are enumeration problems for hypergraph properties α(X) and β(X), respectively (where X is a nonempty subset of vertices in the given hypergraph), then P is at least as hard as Q if an output-polynomial time algorithm for Q implies an output-polynomial time algorithm for P ; and P is equivalent to Q if each of the two enumeration problems is at least as hard as the other.
The equivalence considered in this article (Section 3), while being prompted by the one in [6], focuses on behaviour of solutions instead of their enumeration.So, the problems HY P − T RAN S − H and HY P − DOM − H have been viewed not as enumeration problems but, rather, as problems admitting common solutions under specific conditions.

Trim Hypergraphs
Proof.If | V | = 1 then the conclusion is obvious.In the case | V |> 1, let x, y ∈ V be distinct.Then V − {y} is not a transversal, and so {y} is a hyperedge in H.

Resume
The only class of loop-free Sperner hypergraphs in which the HYP-TRANS and HYP-DOM problems are solution-wise equivalent is the class of trim hypergraphs.
Consequently, enumerating the minimal transversals in a trim hypergraph H is equivalent to enumerating the minimal dominating sets in H.A fortiori, an output-polynomial time algorithm for HYP-TRANS-H is one for HYP-DOM-H, and vice-versa.A future direction of research could be investigation of output-polynomial time algorithm to enumerate minimal transversals / minimal dominating sets in trim hypergraphs.

3. 1 :
Proposition.If H is trim, then HY P − T RAN S − H(≡)HY P − DOM − H. Proof.Let Y ∈ HY P − T RAN S − H. Then Y is a proper subset of V (by 2.4) and Y is a dominating set (by 2.1).Were some proper subset X of Y a dominating set in H, then X would be a transversal (by 2.3), contradicting Y ∈ HY P − T RAN S − H.So Y is a minimal dominating set in H, whence HY P − T RAN S − H ≺ HY P − DOM − H. On the other hand, let Y ∈ HY P − DOM − H. Then Y is a transversal in H (by 2.3).Were some proper subset X of Y a transversal in H, then X would be a dominating set (by 2.1), going against Y ∈ HY P − DOM − H.So Y is a minimal transversal in H, giving HY P − DOM − H ≺ HY P − T RAN S − H as well.Proposition 3.2.If HY P − T RAN S − H(≡)HY P − DOM − H then H is trim.Proof.Let D be a given dominating set in H. Let D 1 be any minimal dominating set contained in D.Then, by hypothesis, D 1 is a minimal transversal in H and so D is a transversal in H. Then H is trim (by 2.3).Propositions 3.1 and 3.2 imply the following proposition.Proposition 3.3.HY P − T RAN S − H(≡)HY P − DOM − H if and only if H is trim.
[7]as no redundant hyperedges then H is a trim hypergraph.Evidently a trim hypergraph is Sperner, though not conversely.Trim hypergraphs are dealt with in some detail in[7].Proposition 2.1.Every transversal is a dominating set, in any hypergraph.Proposition 2.2.In a trim hypergraph H, a set X ∈ 2 V * is a dominating set if and only if X is a transversal.Proposition 2.3.A Sperner hypergraph H is trim if and only if every dominating set is a transversal.