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Separation Dimension and Sparsity

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Separation Dimension and Sparsity Separation Dimension and Sparsity Noga Alon1, Manu Basavaraju2, L. Sunil Chandran3, Rogers Mathew4, and Deepak Rajendraprasad5 1 Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. [email protected] 2 Department of Computer Science and Engineering, National Institute of Technology Karnataka, India. [email protected] 3 Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India. [email protected] 4 Department of Computer Science, Indian Institute of Technology Kharagpur, India. [email protected] 5 Indian Institute of Technology Palakkad, India. [email protected] December 6, 2017 Abstract The separation dimension π(G) of a hypergraph G is the smallest natural number k k for which the vertices of G can be embedded in R so that any pair of disjoint edges in G can be separated by a hyperplane normal to one of the axes. Equivalently, it is the cardinality of a smallest family F of total orders of V (G), such that for any two disjoint edges of G, there exists at least one total order in F in which all the vertices in one edge precede those in the other. Separation dimension is a monotone parameter; adding more edges cannot reduce the separation dimension of a hypergraph. In this article we discuss the influence of separation dimension and edge-density of a graph on one another. On one hand, we show that the maximum separation dimension of a k-degenerate graph on n vertices is O (k lg lg n) and that there exists a family of 2-degenerate graphs with separation dimension Ω (lg lg n). On the other hand, we show that graphs with bounded separation dimension cannot be very dense. Quantitatively, we prove that n-vertex graphs with separation dimension s have at most 3(4 lg n)s−2 · n edges. We do not believe that this bound is optimal and give a question and a remark on the optimal bound. Keywords: separation dimension, edge density, degeneracy. 1 Introduction Let σ : U ! [n] be a permutation of elements of an n-set U and let ≺σ denote the associated total order. For two disjoint subsets A; B of U, we say A ≺σ B when every element of A precedes every element of B in σ, i.e., σ(a) < σ(b); 8(a; b) 2 A × B. We say that σ separates A and B if either A ≺σ B or B ≺σ A. For two subsets A; B of U, we say A σ B when A n B ≺σ A \ B ≺σ B n A. 1 Definition 1.1. A family F of permutations of V (H) is pairwise suitable for a hypergraph H if, for every two disjoint edges e; f 2 E(H), there exists a permutation σ 2 F which separates e and f. The cardinality of a smallest family of permutations that is pairwise suitable for H is the separation dimension of H and is denoted by π(H). A family F = fσ1; : : : ; σkg of permutations of a set V can be seen as an embedding of V k into R with the i-th coordinate of v 2 V being σi(v). Similarly, given any embedding of V in Rk, we can construct k permutations by projecting the points onto each of the k axes and then reading them along each axis, breaking any ties arbitrarily. From this, it is easy to see that π(H) is the smallest natural number k so that the vertices of H can be embedded into Rk such that any two disjoint edges of H can be separated by a hyperplane normal to one of the axes. This motivates us to call such an embedding a separating embedding of H and π(H) the separation dimension of H. The notion of separation dimension was introduced by the authors in [2]1 and further studied in [1, 4]. Apart from its naturalness, a major motivation to study this notion of separation is its interesting connection with a certain well studied geometric representation of graphs. The boxicity of a graph G is the minimum natural number k for which G can be represented as an intersection graph of axis-parallel boxes in Rk. The separation dimension of a hypergraph H is equal to the boxicity of the intersection graph of the edge set of H, i.e., the line graph of H [4]. 1.1 Related notions Families of permutations which satisfy some type of “separation” properties have been long studied in combinatorics. One of the early examples is the work of Ben Dushnik in 1950 where he introduced the notion of k-suitability [6]. A family F of permutations of [n] is k-suitable if, for every (k − 1)-set A ⊆ [n] and for every b 2 [n] n A, there exists a σ 2 F such that A ≺σ fbg. Let N(n; k) denote the cardinality of a smallest family of permutations that is k-suitable for [n]. In 1972, Spencer [15] proved that lg lg n ≤ N(n; 3) ≤ N(n; k) ≤ k2k lg lg n. Fishburn and Trotter, in 1992, defined the dimension of a hypergraph on the vertex set [n] to be the minimum size of a family F of permutations of [n] such that every edge of the hypergraph is an intersection of initial segments of F [8]. It is easy to see that an edge e is an intersection of initial segments of F if and only if for every v 2 [n] n e, there exists a permutation σ 2 F such that e ≺σ fvg. It is interesting to note that, according to this definition, N(n; k) is the dimension of the complete (k − 1)-uniform hypergraph on n vertices. The quantity N(n; 3), which is thus the dimension of the n-vertex complete graph, and which is also the (poset) dimension of the inclusion poset of 1 and 2 sized subsets of [n] is a parameter that appears in many combinatorial problems, including the one we study here. Very tight estimates which can determine the exact value of N(n; 3) for almost all n were given by Hoşten and Morris in 1999 by finding a nice equivalence of this problem to a variant of the Dedekind problem [12]. Füredi, in 1996, studied the notion of 3-mixing family of permutations [10]. A family F of permutations of [n] is called 3-mixing if for every 3-set fa; b; cg ⊆ [n] and a designated element a in that set, one of the permutations in F places the element a between b and c. It is clear that a is between b and c in a permutation σ if and only if fa; bg σ fa; cg or fa; cg σ fa; bg. Such families of permutations with small sizes have found applications in showing upper bounds for many combinatorial parameters including poset dimension [13], product dimension [9] and boxicity [5]. 1 Some of our initial results on this topic including some of the results reported here are available as a preprint in arXiv [3]. 2 1.2 Our results The separation dimension of Kn, the complete graph on n vertices is Θ (lg n) [4]. It is easy to see that separation dimension is a monotone property, i.e., π(G0) ≤ π(G) if G0 is a subgraph of G. So it is interesting to check whether the separation dimension of sparse graph families can be much lower than lg n. Here, by a sparse family of graphs, we mean a family of graphs in which the number of edges is linear in the number of vertices. Sparsity of a graph family, as we consider it here, is equivalent to the restriction that the graphs in the family have bounded average degree. It is easy to see that globally sparse graphs with a small dense subgraph can have large separation dimension.p For example if we consider an n vertex graph which is a disjoint union of a complete graph on b nc vertices and remaining isolated vertices, it has at most n=2 edges, but has a separation dimension in Ω (lg n) due to the clique. Hence sparsity is needed across all subgraphs in order to hope for a better upper bound on separation dimension. One common way of ensuring local sparsity of a graph family is to demand that the degeneracy of the graphs in the family be bounded. Definition 1.2 (Degeneracy). For a non-negative integer k, a graph G is k-degenerate if the vertices of G can be linearly ordered in such a way that every vertex is succeeded by at most k of its neighbours. The least number k such that G is k-degenerate is called the degeneracy of G and any such enumeration is referred to as a degeneracy order of V (G). For example, trees and forests are 1-degenerate and planar graphs are 5-degenerate. Series- parallel and outerplanar graphs are 2-degenerate. Graphs of treewidth t are t-degenerate. It is easy to verify that if the maximum average degree over all subgraphs of a graph G is d, then G is bdc-degenerate. A bdc-degeneracy order of G can be obtained by recursively picking out a minimum degree vertex from G. It is also easy to see that any subgraph of a k-degenerate graph has average degree at most 2k. In this paper we establish the following upper bound on the separation dimension of k- degenerate graphs and thereby give an affirmative answer to our question under a restricted but necessary condition of sparsity.
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