On the P3-hull number of Kneser graphs

This paper considers an infection spreading in a graph; a vertex gets infected if at least two of its neighbors are infected. The P3-hull number is the minimum size of a vertex set that eventually infects the whole graph. In the specific case of the Kneser graph K(n, k), with n > 2k + 1, an infection spreading on the family of k-sets of an n-set is considered. A set is infected whenever two sets disjoint from it are infected. We compute the exact value of the P3-hull number of K(n, k) for n > 2k + 1. For n = 2k + 1, using graph homomorphisms from the Knesser graph to the Hypercube, we give lower and upper bounds. Mathematics Subject Classifications: 05C76, 52A37, 05C85


Introduction
We only consider finite, simple, and undirected graphs. For a graph G = (V, E), a graph convexity on V is a collection C of subsets of V such that ∅, V ∈ C and C is closed under intersections. The sets in C are called convex sets and the convex hull H C (S) in C of a set S of vertices of G is the smallest set in C containing S (see [7] and references therein). Some natural convexities in graphs are defined by a set P of paths in G, in a way that a set S of vertices of G is convex if and only if for every path P : v 0 , v 1 , . . . , v l ∈ P such that v 0 and v l belong to S, all vertices of P belong to S (cf. [1,8]). If we define P as the set of all shortest paths in G, we have the well-known geodetic convexity (see for example [17,11,25]). The monophonic convexity is defined by considering P as the set of all induced paths of G [18,15].
If we let P be the set of all paths of G with three vertices, we have the well-known P 3 -convexity which will be studied in this paper. This convexity was introduced with the aim of modeling the spread of a disease in a grid [5]. Since then, many articles, in connection with this convexity, were published in the specialized literature (the reader is referred for instance to [10,9,2,16,7]).
Given a set S ⊆ V , the P 3 -interval I[S] of S is formed by S, together with every vertex outside S with at least two neighbors in S. If I[S] = S, then the set S is P 3 -convex. The P 3 -convex hull H C (S) of S is the smallest P 3 -convex set containing S. In what follows, we write H(S) instead of H C (S). The P 3 -convex hull H(S) can be formed from the sequence , for all q p, then I p [S] is a P 3 -convex set. If H(S) = V (G) we say that S is a P 3 -hull set of G. The cardinality h P 3 (G) of a minimum P 3 -hull set in G is called the P 3 -hull number of G. Centeno et al. proved that, given a graph G and an integer k, to decide whether the P 3 -hull number of G is at most k is an NP-complete problem [10]. Coelho et al. [14] proved that compute the P 3 -hull number is an APX-hard problem even for bipartite graphs with maximum degree four. Moreover, Chen [12] shown that the P 3 -hull number of a graph is hard to approximate within a ratio O(2 log 1− n ), for any > 0, unless NP ⊆ DTIME(n polylog(n) ). All these negative results motivate the study of the P 3 -hull number on particular families of graphs.
In this paper we deal with the problem of computing the P 3 -hull number of Kneser graphs K(n, k). Kneser graphs have a very nice structure. For an overview on this relevant family of graphs we refer the reader to [20]. Many graph theoretic parameters have been computed for Kneser graphs K(n, k). Some examples are the independence number [19], the chromatic number [22], the diameter [28].
The aim of this work is twofold, first to contribute to the knowledge of Kneser graphs; second to obtain new formulas for the hull number within a family of graphs having nice structure.
This article is organized as follows. In Section 2 we present some preliminaries definitions and concepts. Section 3 is devoted to our results. Finally, we give some concluding remarks in Section 4.

Related work
Infection problems appear in the literature under many different names and were studied by researches of various fields [13]. An infection problem already studied on Kneser graphs is zero forcing (see [6]). The zero forcing problem follows the infection rule where an infected vertex v will infect one of its neighbors w if all neighbors of v except for w are already infected. The zero forcing number of G is the size of a smallest set S of initially infected vertices that forces the whole graph to become infected. Another infection problem is the bootstrap percolation on a graph (see for example, [4,3,23,24,26,27] and references therein): an infection spreads over the vertices of a connected graph G following a deterministic spreading rule in such a way that an infected vertex will remain infected forever. Given a set S ⊆ V (G) of initially infected vertices, we can build a sequence S 0 = S, S 1 , S 2 , . . . in which S i+1 is obtained from S i using such a spreading rule. Under the r-neighbor bootstrap percolation on a graph G, the spreading rule is a threshold rule in which S i+1 is obtained from S i by adding to it the vertices of G which have at least r neighbors in S i . The set S 0 is a percolating set of G if there exists a t such that S t = V (G). Let t r (S) be the minimum t such that S t = V (G). The percolation time of G is defined as t r (G) = max{t r (S) : S percolates G}. Notice that this infection problem is related to graph convexities. In fact, the 2-neighbor bootstrap percolation problem on graphs is very close to the P 3 -convexity on graphs. The 2-neighbor bootstrap percolation problem has been studied by several authors. For example, the maximum percolation time of the 2-neighbor bootstrap percolation problem has been studied by Benevides et al. [4], Marcilon et al. [23] and Przykucki [26]. The smallest or largest size of a percolating set with a given property has been studied by Benevides et al. [3] and Morris [24]. Moreover, Przykucki [26] and Riedl [27] studied some problems concerning the size of 2-percolating sets. Notice that the problem of finding a minimum size 2-percolating set on a graph is equivalent to determining the P 3 -hull number of such graph. As we have mentioned previously, the problem of computing the P 3 -hull number of a graph is a very hard problem, even for bipartite graphs. Therefore, it is interesting to find infinite graph families where such parameter can be easily determined in polynomial time.

Preliminaries
Given a graph G, N G (u) stands for the neighborhood of u in G. Let A and B be two sets. Given an integer a such that 0 a |A|, A a stands for the set whose elements are the a-element subsets of A, and A a B b the set whose elements are the subsets of A ∪ B with a elements in A and b elements in B.
Let n be a positive integer. We denote by [n] the set {1, · · · , n}. For positive integers n and k such that n 2k, the Kneser graph, denoted K(n, k), has as vertex set [n] k and two vertices are adjacent if they have empty intersection.
We introduce two more graphs in order to study the P 3 -hull number of the Kneser graph K(2k+1, k), the n-cube and middle levels graph. For any n ∈ Z + , the n-dimensional hypercube (or n-cube), denoted Q n , is the graph in which the vertices are all binary ntuples of length n (i.e., the set {0, 1} n ), and two vertices are adjacent if and only if they differ in exactly one position. For any i ∈ {0, . . . , n} we denote by Q i n the ith-layer of Q n , that is, the subgraph of Q n induced by all the vertices having exactly i ones.
The middle levels graph M 2k+1 is the graph whose vertices are all k-element and all (k + 1)-element subsets of {1, 2, . . . , 2k + 1}, with an edge between any pair of sets where one is a proper subset of the other. The name middle levels graph for M 2k+1 comes from the fact that it is isomorphic to the subgraph of the hypercube Q 2k+1 induced by all the vertices in the middle two layers Q k 2k+1 and Q k+1 2k+1 . It is not difficult to see that M 2k+1 is a bipartite connected graph of order 2 2k+1 k . Johnson and Kierstead [21] provide a natural 2-to-1 graph homomorphism φ from M 2k+1 to K(2k + 1, k) defined by:

Hull number of Kneser graphs
Let k 1 and n 2k + 1. Proof. First we show by induction that F t ∪ F k−t ⊂ H(F 1 ) for t = 1, . . . , k/2 . To do this notice that taking i = j = k − 1 in Lemma 1 we obtain the base case t = 1. Now assume the statement is true for t 1. Taking i = t + 1 and j = t in Lemma 1 we obtain completing the induction. To finish the proof, notice that taking i = k and j = k − 1 in Lemma 1 we obtain F k ⊂ I[F 1 ] and taking i = j = 0 in Lemma 1 we obtain F 0 ⊂ I[F k ].
Notice that A 1 and A 2 are neighbors of all the vertices in F 0 . Hence  Proof. First, we will prove that h P 3 (K(2k + 2, k)) > 2. Let S = {S 1 , S 2 } ⊆ K(2k + 2, k) and let A = S 1 ∪ S 2 . We split the proof into the only two possible cases for |S 1 ∩ S 2 |. Since |A| = k + 1, each vertex in A k is adjacent to S 1 and S 2 and thus A k ⊆ I[S]. Symmetrically, since |A| = k + 1 and A k ⊆ I[S], we conclude that A k ⊆ I 2 [S]. Let C be any vertex in K(2k + 2, k). Since |C| = k 3, either |C ∩ A| 2 or |C ∩ A| 2. Assume, without loosing generality, that |C ∩ A| 2. If C / ∈ A k ∪ A k , then |C ∩ A| 1. Hence C has no neighbors in A k and it has at most one neighbor in A k which implies Let A = S 1 ∪ S 2 . Hence, |A| = k + 2 and |A| = k. Thus, A k = {A} and I[S] = {A, S 1 , S 2 }. In addition, for each i ∈ {1, 2}, C ∩ S i = ∅ for every C ∈ A k . Therefore, H(S) = {S 1 , S 2 , A}.

Since in both cases
]\{k} and A 3 = {3, . . . , k + 2}. We will prove that S is a hull set of K(2k + 2, k). As in the proof of Theorem 3, A 1 and A 2 are neighbors of all the vertices in F 0 and hence F 0 ⊂ Now we claim that F 1 ⊂ N (S 2 ). This implies that F 1 ⊂ H(S), as S 2 ∩ F k ⊂ F k−1 ∩ F k = ∅ and F 1 ⊂ N (F k ). Notice that by Lemma 2 we obtain H(S) = K(2k+1, k). Remark 5. h P 3 (K(6, 2)) = 2.
Hence C is adjacent to A \ C ∈ A 2 and A \ C ∈ A 2 and thus C ∈ H({S 1 , S 2 }). Therefore, {S 1 , S 2 } is a hull set of K(6, 2).

Preservation of P 3 convexity under homomorphisms and its inverses
Let G = (V, E) a graph. For any vertex u ∈ V , let N G (u) denote the subset of neighbor vertices of u in G, that is, the set {v ∈ V : uv ∈ E}. Let G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) be graphs. A graph homomorphism between graphs G 1 and G 2 , denoted by φ : G 1 → G 2 , is a mapping φ from V (G 1 ) to V (G 2 ) such that φ(u) and φ(v) are adjacent in G 2 whenever u and v are adjacent in G 1 . A graph homomorphism φ : G 1 → G 2 is called locally bijective if for all u ∈ G 1 the restriction of φ to N G 1 (u) is a bijection between N G 1 (u) and N G 2 (φ(u)).

Proof. We prove by induction on
In the base case i = 0 we actually have equality. Now assume the statement is true for i > 0.
Then there are two neighbors v and w of φ(u) in I i (S). By assumption,  Theorem 9. Let φ : G 1 → G 2 be a surjective, locally bijective graph homomorphism.
Proof. The result follows from Theorem 9 by noticing that the 2-to-1 graph homomorphism from M 2k+1 to K(2k + 1, k) defined at the end of Section 2 is surjective and locally bijective.

Lower bound of h P 3 (K(2k + 1, k))
In order to deduce a lower bound for h P 3 (K(2k + 1, k)), we need the following preliminary results.
Lemma 11. Let n > 1 and 1 i n − 1 be integers. Let S be the set of vertices in the ith-layer Q i n of the hypercube Q n . Then, S is a P 3 -hull set of Q n . Proof. Let x = (x 1 , · · · , x n ) be any vertex in Q i−1 n . Clearly, there exist two coordinates x p , x q in x, with 1 p < q n, such that x p = x q = 0. The vertices y = (x 1 , · · · , x p−1 , 1, x p+1 , · · · , x n ) and z = (x 1 , · · · , x q−1 , 1, x q+1 , · · · , x n ) are vertices in S adjacent to x. In the same way, for any vertex w in Q i+1 n we can pick two different coordinates w p and w q such that w p = w q = 1. Then we can find two vertices u and v in S adjacent to w, where u (resp. v) is equal to w except in the pth (resp. qth) coordinate which is equal to 0. Thus, w has at least two neighbors in S. As this property holds for any 1 i n − 1 then, we conclude that S is a P 3 -hull set of the hypercube Q n .
Proof. Let S be a P 3 -hull set of M 2k+1 . For any vertex w ∈ S letw be a vertex in the hypercube Q 2k+1 such thatw j = 1 if j ∈ w, andw j = 0 otherwise, for 1 j 2k + 1. As M 2k+1 is isomorphic to the subgraph of Q 2k+1 induced by the vertices in the two middle layers Q k 2k+1 and Q k+1 2k+1 then, by Lemma 11, the set S = {w : w ∈ S} is a P 3 -hull set of Q 2k+1 . Therefore, by Theorem 12, |S | 2k+1 2 Finally, by Lemma 13 and Corollary 10, we have the following theorem.

Discussion
Corollary 10 gives an upper bound for the P 3 -hull number of M 2k+1 in terms of the P 3 -hull number of K(2k + 1, k). Exact values for h P 3 (K(2k + 1, k)) and h P 3 (M 2k+1 ), calculated with the aid of a computer, are shown in Table 1. So we have the following conjecture.
The lower bound for the P 3 -hull number of K(2k + 1, k) obtained in Theorem 14 seems to be far from being tight. In addition to results given in Table 1, we also have computational evidence showing that h P 3 (K(2k + 1, k)) is at most equal to 11, 16 and 23 for k = 5, 6 and 7, respectively. Notice that h P 3 (K(2k + 1, k)) seems to be equal to k(k−1) 2 + c, being c a constant, with c 2. So we have the following conjecture.