INDEPENDENT DOMINATION IN THE GRAPH DEFINED BY TWO

. Fix a positive integer n and consider the bipartite graph whose vertices are the 3-element subsets and the 2-element subsets of [ n ] = { 1 , 2 ,...,n } , and there is an edge between A and B if A ⊂ B . We prove that the domination number of this graph is (cid:0) n 2 (cid:1) − ⌊ ( n +1) 2 8 ⌋ , we characterize the dominating sets of minimum size, and we observe that the minimum size dominating set can e chosen as an independent set. This is an exact version of an asymptotic result from [3]. For the corresponding bipartite graph between the ( k + 1)-element subsets and the k -elements subsets of [ n ] ( k > 3), we provide a new construction for small independent dominating sets. This improves on a construction from [5], where these independent dominating sets have been studied under the name saturating ﬂat antichains.


Introduction
For a graph G = (V, E), a vertex set D ⊆ V is called dominating if every vertex v ∈ V \ W has a neighbor in W .The domination number γ(G) is the minimum size of a dominating set in G. Adding the condition that the set D is an independent set in G, that is, no two vertices in D are adjacent, we obtain the independent domination number, denoted by i(G).These graph parameters have been studied intensively [6,9,10].Clearly, γ(G) i(G), and it is known that γ(G) = i(G) for claw-free graphs G [1].
Let n be a positive integer.We use [n] to denote the set {1, 2, . . ., n}, and for a positive integer k < n, [n]   k denotes the set of k-element subsets of [n].For 1 k < l n, let G l,k be the bipartite graph with vertex set V = [n]  k ∪ [n]   l where A ∈ [n]   k and B ∈ [n]   l are adjacent if A ⊆ B. The investigation of the domination number of G l,k has been initiated in [2,3].In this paper we focus on the case l = k + 1.For (l, k) = (3, 2) we extend the arguments from [8] to confirm a conjecture from [2], an asymptotic version of which has been established in [3] (for general (l, 2)).
In the proof, it turns out to be useful to associate with a dominating set D = D 2 ∪ D 3 where D i = D ∩ [n]   i another graph H = H(D) with vertex set V (H) = [n] and edge set E(H) = ∆D 3 \ D 2 where ∆D 3 = {A : |A| = 2 and A ⊆ X for some X ∈ D 3 }.From D being a dominating set it follows that the vertex set of every triangle of the graph H is an element of D 3 .Moreover, if every edge of H is contained in a triangle in H (in other words, for every edge xy, the vertices x and y have a common neighbor z), and D is a minimal dominating set, then D 3 is the set of triangles in H and D 2 is the set of non-edges: 2 \ E(H).In this situation, D is an independent set in G 3,2 .
We will show that the graphs coming from minimum dominating sets in G 3,2 can be described as follows.For 1 < s < n, let K + s,n−s be the graph such that [s] ⊆ [n] induces a matching of size ⌊s/2⌋ (and an additional isolated vertex if s is odd), [s + 1, n] is an independent set, and uv ∈ E K + s,n−s for all u ∈ [s], v ∈ [s + 1, n] (see Figure 1).Moreover, let H 5a , H 5b and H 9 be the three graphs shown in Figure 2.
All but one of the graphs in Theorem 2 have the property that every edge is contained in a triangle, so that the corresponding dominating set D is determined by the graph H, and is an independent set in G 3,2 .The exception is the graph K + (n+1)/2,(n−1)/2 for n ≡ 1 (mod 4).In this graph the edges that are not contained in a triangle form a star S, say with center v, with n−1 2 leaves, and a corresponding minimum dominating set in G 3,2 is obtained by taking D 2 to be the set of non-edges and D 3 the set of triangles together with n−1 4 sets of the form {v, x, y} where x and y are leaves of S (see the proof of Theorem 2 for a precise statement).
For k 3, [5, Theorem 14] bounds the minimum size of an independent dominating set in G k+1,k as follows: Here, the t k is the Turán-density for the complete k-uniform hypergraph on k + 1 vertices, that is, , where ex(n, K k+1 ) is the maximal number of edges in a k-uniform hypergraph on n vertices without a complete k-uniform hypergraph on k + 1 vertices as a subhypergraph.We improve the upper bound in (1) as follows.
Theorem 3.For every k 3 and every α with 0 < α < 1, This bound is minimized for α being the root of the polynomial (k − 1)x k − kx + 1 = 0 in the interval [0, 1  2 ].For small k, the values for the bounds are collected in Table 1, where we used the upper bounds for t k from [4] for odd k and [12] for even k.For a set X of vertices in a graph G = (V, E), G[X] denotes the subgraph induced by X and E(X) denotes the set of edges with both endpoints in X.Similarly, for X, Y ⊆ V , E(X, Y ) is the set of edges with one endpoint in X and one endpoint in Y .For a set A, its shadow ∆A is defined as ∆A = {B : B ⊆ A and |B| = |A| − 1}, and for a family F of k-sets, ∆F = A∈F ∆A = {B : |B| = k − 1 and B ⊆ A for some A ∈ F }.

The domination number of G 3,2
In this section we prove Theorems 1 and 2. The statements for independent domination are just a rewording of the main results in [8], and our approach is to adjust the arguments from this paper so that they apply without the independence assumption.The proof in [8] is based on the observation that there is a one-to-one correspondence between independent dominating sets in G 3,2 and graphs with the property that every edge is contained in a triangle.More precisely, given such a graph H the dominating set in G 3,2 consists of the 3-sets which are triangles in H and the 2-sets which are not edges in H. Conversely, for a given independent dominating set D = D 2 ∪ D 3 , we obtain the associated graph H by setting E(H) = ∆D 3 .Dropping the independence assumption, we still define a graph associated with a dominating set as follows.
The graph H has not necessarily the property that every edge is contained in a triangle, and going from D to H(D) we are losing some information: in general, it is not possible to reconstruct D from H. Nevertheless, we can bound |D| in terms of H and this bound turns out to be sufficient to prove Theorems 1 and 2.
Lemma 2.1.Let H = (V, E) be the graph associated with a minimal dominating set D = D 2 ∪ D 3 .Let T be the set of triangles in H, and let E 0 be the set of edges that are not contained in a triangle.Then Moreover, for every xy ∈ E 0 there must be a 3-set in D 3 \ T containing x and y.Since any such 3-set covers at most 2 elements of E 0 , we obtain As a consequence, an upper bound on |E| − |T | − is immediate by looking at the dominating sets corresponding to the graphs listed in Theorem 2.
Lemma 2.2.Let H = (V, E) be a graph with vertex set V = [n].Let T be the set of triangles in H, and let E 0 be the set of edges that are not contained in a triangle.Then Moreover, equality is possible only if Proof.Fix a triangle xyz ∈ T .Counting the edges between {x, y, z} and V \ {x, y, z} and the triangles with two vertices in {x, y, z}, we find where α xyz 0 is the number of vertices in V \ {x, y, z} which have either 0 or 3 neighbors in {x, y, z}.Taking the sum over T and setting α and subtracting xy∈E (t(xy For every xy As a consequence, in the second sum on the left-hand side of (3), only the terms for xy ∈ E 0 can be negative, and it follows that where (with E 1 = E \ E 0 being the set of edges contained in at least one triangle) Next we substitute into (4) and rearrange the result: is an integer, hence (2) follows with , and with ) follows from ( 5), and follows by rounding (5).If E 0 = ∅, then the right-hand side of ( 5) is strictly less than , and (2) follows with a strict inequality if |E 0 | is odd.Theorem 1 is an immediate consequence of Lemmas 2.1 and 2.2.To prove Theorem 2, we start with the observation that the optimal independent dominating sets have been described completely in [8].It remains to check whether there are any additional dominating sets of size γ(G 3,2 ), such that E 0 = ∅ in the associated graph.
Proof of Theorem 2. .Let H be the graph associated with D. For n ≡ 1 (mod 4), E 0 = ∅, so that D is an independent set, and the result follows from [8].It remains the case n ≡ 1 (mod 4).That this might get a bit more complicated is indicated by the observation that this is the only case where one of the graphs listed in Theorem 2 satisfies E 0 = ∅: For H = K (n+1)/2,(n−1)/2 there is one isolated vertex, say vertex 1, in the subgraph induced by 1, 2, . . ., (n + 1)/2, and in H this vertex is adjacent to every vertex in the independent set {(n + 3)/2, . . ., n}.So E 0 = {1x : x = (n + 3)/2, . . ., n}, and an optimal dominating set corresponding to H is given by Our proof that the list for n ≡ 1 (mod 4) is complete follows closely the corresponding proof in [8].Some small modifications are needed to take into account the possibility that E 0 = ∅.The case n = 5 can be treated by hand, and from now on we assume n > 9.In the following, we will formulate a sequence of claims providing more and more information about the structure of H, eventually allowing us to show that H must be isomorphic to one of the graphs listed in the theorem.To keep the presentation of our main argument reasonably short, we postpone the proofs of the claims (some of which are a bit tedious) to Appendix A. In view of (5) we start from Let M be the set of edges defined by From now on, we assume 4 then H is isomorphic to K + (n−1)/2,(n+1)/2 or K + (n+1)/2,(n−1)/2 .From now on we assume |M | n−5 4 , and introduce the following notation: We also set ] contains at most one isolated vertex and no isolated edge.In particular, Claim 12. δ n−1 2 − 2. Finally, we can derive the required contradiction.By Claim 12, there are x, y 2 and let H ′ = H − x be the graph obtained from H by deleting vertex x.With E ′ , T ′ and E ′ 0 being the sets of edges, triangles, and edges not contained in a triangle in H ′ , we have Using Claim 11 and our assumption |M | n−5 4 , we obtain | is an integer, so we can round the right-hand side (taking into account that n ≡ 1 (mod 4)): and this implies /2 the neighborhood of any vertex of degree n+1 2 induces a star K 1,(n−1)/2 .
3. An upper bound for i(G k+1,k ) In this section we prove Theorem 3.For k = 2, the independent dominating sets in G k+1,k correspond to graphs with the property that every edge is contained in a triangle.We generalize this to k-uniform hypergraphs (k-graphs for short) in the following way.Definition 2. A k-graph is well-covered if every edge is contained in a (k + 1)-clique.In other words, for every edge e = {v 1 , . . ., v k }, there exists a vertex v k+1 ∈ V (H) \ e such that e i = e \ {v i } ∪ {v k+1 } is an edge for every i ∈ As in the case k = 2, there is a correspondence between well-covered k-graphs H and independent dominating sets D in G k+1,k .The dominating set D(H) has the (k + 1)-cliques of H as the vertices on level k + 1, and the k-sets which are not edges of H as the vertices on level k.Conversely, given an independent dominating set D in G k+1,k we get a well-covered k-graph H(D) by taking the k-sets which are not in D as the edges.For a k-graph H, let e(H) and c(H) be the numbers of edges and of (k + 1)-cliques in H, respectively.The above observation implies that finding i(G k+1,k ) is equivalent to maximizing e(H) − c(H) over all well-covered k-graphs H.More precisely, i(G k+1,k ) = n k − e(H) + c(H) for an optimal H.Our approach is to generalize the graphs K + s,n−s (for even s) from Theorem 2. One way of looking at this construction is as follows: In order to construct a graph which has many edges and few triangles subject to the condition that every edge is contained in a triangle, we partition the vertex set into two parts A and B, take all edges with exactly one vertex in B (and one in A), and add a set M of edges in A with the property that every x ∈ A is contained in exactly one edge in M (that is, a perfect matching on A).Translating this directly to k-graphs, we would like to take all k-sets with exactly one vertex in B (and k − 1 vertices in A), and then add a collection M of k-subsets of A such that every (k − 1)-subset of A is contained in exactly one member of M .The complete k-graphs on k + 1 vertices are then precisely the sets X ∪ {b} with X ∈ M and b ∈ B. As the design condition on M (every (k − 1)-set is covered exactly once) is too much to hope for we modify the requirement as follows: We take M to be a collection of k-subsets of A such that every (k − 1)-subset of A at most once.As a consequence, not all k-sets with exactly one vertex in B can be edges, but only those of the form X ∪ {b} with b ∈ B and X a (k − 1)-subset of A that is covered by a member of M .In the following lemma we determine a lower bound for the value e(H) − c(H) that can be obtained using this construction.
In contrast to the case k = 2, for k 3 the construction can still be improved.Note that so far the set B is very thin: It contains at most 1 vertex from every edge.As a consequence, we can add another k-graph H ′ on the vertex set B, and the graph H ′′ obtained by taking the union of the edge sets of H and H ′ satisfies e(H ′′ ) − c(H ′′ ) = e(H) − c(H) + e(H ′ ) − c(H ′ ).A natural idea is to use the same construction for H ′ as for H, and then this can be iterated.Now we want to make the above idea more precise.We apply the construction from Lemma 3.1 recursively.

Start with a partition [n] = A
For i ∈ {0, 1, . . ., r − 1}, let H i be the k-graph from Lemma 3.1 on the vertex set and let H be the k-graph on the vertex set [n] with edge set In the following lemma we provide the asymptotics for e(H) − c(H).
Proof.Fix ε > 0 and then fix ε 1 > 0 satisfying Choose N sufficiently large such that Hence, if n is sufficiently large, then for all i ∈ {0, 1, . . ., N }, Taking the sum over i, we obtain and this concludes the proof.
Maximizing the right hand side in Lemma 3.2 , we choose α to be the root of the polynomial (k−1)x k −kx+1 in the interval [0, 1/2], and substituting kα and this concludes the proof of Theorem 3.

Conclusion and open problems
In this paper, we have investigated the independent domination number of G k+1,k .For k = 2, we proved the exact value and a complete characterization of the smallest independent dominating sets, and for k 3 we improved the asymptotic upper bound from [5].It would be nice to close the gap between the upper and lower bound, so we restate the following problem from [5] in our notation.
Problem 1. Try to close the gap between the lower bound in (1) and the upper bound in Theorem 3. In particular, find a lower bound that does not depend on hypergraph Turán densities.
Another natural direction for further investigations is to consider G l,k where l and k are not consecutive.It is not hard to see that the lower bound in (1) actually applies to γ(G k+1,k ) instead of i(G k+1,k ) and generalizes as follows: One problem with this bound is that the exact values of t l,k are not known for k 3.So it makes sense to focus on the case k = 2 with t l,2 = l−2 l−1 .In this case it has been proved that the above lower bound for γ(G l,2 ) is tight [3]: For l = 3 the smallest dominating set can be taken to be independent, but for l 4 it looks plausible that i(G l,2 ) > γ(G l,2 ).An asymptotic variant would be the following.
Problem 2. Prove that for every l 4, there exists ε > 0 with For instance, for l = 4, the best construction we are aware of is from [11] and gives i(G 4,2 ) Appendix A. Proofs for the claims in the proof of Theorem 2 Recall that we assume n ≡ 1 (mod 4) and n > 9, and that for n ≡ 1 (mod 4), we have already established that the graphs listed in Theorem Proof.Assume M = ∅.From and ( 6), we obtain Using the optimality of H and Lemma 2.2, From now on, we assume 0 be the set of edges, the set of triangles, and the set of edges not in a triangle, respectively.Then and with , the claim follows.
From now on we assume d(x and then Similarly, for the edges xv, yu and yv, and therefore β n − 3 > n 2 , which contradicts (6).Claim 5. M is an induced matching with |M | n−1 Proof.If xy, yz ∈ M then wz ∈ E(V 1 (xy)) for every w ∈ V 1 (xy) \ {z}, which contradicts Claim 4. This shows that M is a matching.Now suppose xy, zw ∈ M and xz ∈ E. By definition of M , yz, xw, yw ∈ E, and then zw ∈ E(V 1 (xy)), again contradicting Claim 4. Therefore, M is an induced matching.In particular, for For 2 for all xy ∈ M .There must be an edge in V \ V (M ) (otherwise we are in Case 1), but there can't be more than one edge because the sets V 1 (xy) are independent sets.So let's say . for every z

From now on we assume
Moreover, for every z ∈ V

Table 1 .
Numerical values of the asymptotic bounds for i(G k,k+1 )/ n k .
Notation.Throughout we will consider only simple graphs.We write xy for the edge {x, y} and xyz for a triangle with vertices x, y and z.For a vertex x, d(x) is the degree of x and N (x) is the neighborhood of x.With t(x) and t(xy) we denote the number of triangles containing the vertex x and the edge xy, respectively.
12 |E 0 | for arbitrary H implies a lower bound for γ(G 3,2 ).Note that we only need the lower bound because γ