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Abstract. Let G be a simple, undirected, finite graph with vertex set V (G) and edge set E(G). A kdimensional box is a Cartesian product of closed intervals [a1, b1] × [a2, b2] × · · · × [a k , b k ]. The boxicity of G, box(G) is the minimum integer k such that G can be represented as the intersection graph of k-dimensional boxes, i.e. each vertex is mapped to a k-dimensional box and two vertices are adjacent in G if and only if their corresponding boxes intersect. Let P = (S, P ) be a poset where S is the ground set and P is a reflexive, anti-symmetric and transitive binary relation on S. The dimension of P, dim(P) is the minimum integer t such that P can be expressed as the intersection of t total orders. Let GP be the underlying comparability graph of P, i.e. S is the vertex set and two vertices are adjacent if and only if they are comparable in P. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset P, box(GP )/(χ(GP ) − 1) ≤ dim(P) ≤ 2box(GP ), where χ(GP ) is the chromatic number of GP and χ(GP ) = 1. It immediately follows that if P is a height-2 poset, then box(GP ) ≤ dim(P) ≤ 2box(GP ) since the underlying comparability graph of a height-2 poset is a bipartite graph. The second result of the paper relates the boxicity of a graph G with a natural partial order associated with the extended double cover of G, denoted as Gc: Note that Gc is a bipartite graph with partite sets A and B which are copies of V (G) such that corresponding to every u ∈ V (G), there are two vertices uA ∈ A and uB ∈ B and {uA, vB} is an edge in Gc if and only if either u = v or u is adjacent to v in G. Let Pc be the natural height-2 poset associated with Gc by making A the set of minimal elements and B the set of maximal elements. We show that box(G)These results have some immediate and significant consequences. The upper bound dim(P) ≤ 2box(GP ) allows us to derive hitherto unknown upper bounds for poset dimension such as dim(P) ≤ 2 tree-width (GP )+ 4, since boxicity of any graph is known to be at most its tree-width + 2. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree ∆ is O(∆ log 2 ∆) which is an improvement over the best known upper bound of ∆ 2 + 2. (2) There exist graphs with boxicity Ω(∆ log ∆). This disproves a conjecture that the boxicity of a graph is O(∆). (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n vertices with a factor of O(n 0.5−ǫ ) for any ǫ > 0, unless N P = ZP P .

A k-dimensional box is the Cartesian product R1 × R2 × · · · × R k where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R1 × R2 × · · · × R k where each Ri is a closed interval on the real line of the form [ai, ai + 1]. The cubicity of G, denoted as cub(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of kcubes. The threshold dimension of a graph G(V, E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm to approximate the threshold dimension of a graph on n vertices with a factor of O(n 0.5−ǫ ) for any ǫ > 0, unless N P = ZP P . From this result we will show that there exists no polynomial-time algorithm to approximate the boxicity and the cubicity of a graph on n vertices with factor O(n 0.5−ǫ ) for any ǫ > 0, unless N P = ZP P . In fact all these hardness results hold even for a highly structured class of graphs namely the split graphs. We will also show that it is NP-complete to determine if a given split graph has boxicity at most 3.

An axis parallel d-dimensional box is the Cartesian product R1 × R2 × · · · × R d where each Ri is a closed interval on the real line.

A k-dimensional box is the cartesian product R 1 × R 2 × · · · × R k where each R i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. Let G be a circular arc graph with maximum degree ∆. We show that if ∆ α but with ∆ = n (α−1) 2α + n 2α(α+1) + (α + 2). So the result cannot be improved substantially when α is large. Let r inf be minimum number of arcs passing through any point on the circle with respect to some circular arc representation of G. We also show that for any circular arc graph G, box(G) ≤ r inf + 1 and this bound is tight. Given a family of arcs F on the circle, the circular cover number L(F ) is the cardinality of the smallest subset F ′ of F such that the arcs in F ′ can cover the circle. Maximum circular cover number L max (G) is defined as the maximum value of L(F ) obtained over all possible family of arcs F that can represent G. We will show that if G is a circular arc graph with L max (G) > 4 then box(G) ≤ 3.