Minimal Embedding Dimensions of Rectangle k-Visibility Graphs

Bar visibility graphs were adopted in the 1980s as a model to represent traces, e.g., on circuit boards and in VLSI chip designs. Two generalizations of bar visibility graphs, rectangle visibility graphs and bar k-visibility graphs, were subsequently introduced. Here, we combine bar kand rectangle visibility graphs to form rectangle k-visibility graphs (RkVGs), and further generalize these to higher dimensions. A graph is a d-dimensional RkVG if and only if it can be represented with vertices as disjoint axis-aligned hyperrectangles in d-space, such that there is an axis-parallel line of sight between two hyperrectangles that intersects at most k other hyperrectangles if and only if there is an edge between the two corresponding vertices. For any graph G and a fixed k, we prove that given enough spatial dimensions, G has a rectangle k-visibility representation, and thus we define the minimal embedding dimension (MED) with k-visibility of G to be the smallest d such that G is a d-dimensional RkVG. We study the properties of MEDs and find upper bounds on the MEDs of various types of graphs. In particular, we find that the k-visibility MED of the complete graph on m vertices Km is at most dm/(2(k + 1))e, of complete r-partite graphs is at most r + 1, and of the m hypercube graph Qm is at most d2m/3e in general, and at most b√m e for k = 0, m 6= 2. E-mail address: espen@slett.net (Espen Slettnes) This work is licensed under the terms of the CC-BY license. 60 E. Slettnes MEDs of RkVGs


Introduction
Bar visibility graphs were introduced in the 1980s as a way to model circuit traces in VLSI chip designs by Lodi and Pagli [16]. A graph G is a bar visibility graph if there is a one-to-one correspondence between its vertices and 2D horizontal bars, such that there is an unobstructed vertical line of sight between two bars (i.e., a vertical line segment between the two bars not intersecting other bars) if and only if there is an edge between the corresponding vertices in G. Note that the bars and visibility lines form a planar graph drawing of G.
In their 1997 paper On Rectangle Visibility Graphs [2], Bose et al. introduced rectangle visibility graphs as "a graph in the plane so that the vertices of the graph are rectangles that are aligned with the axes, and the edges of the graph are horizontal or vertical lines-of-sight". Previously, though using different terminology, Stephen Wismath established in his 1989 thesis [18] that all planar graphs are rectangle visibility graphs (i.e., have rectangle visibility representations).
Dean et al. introduced bar k-visibility graphs in 2007 [6] as a generalization in which visibility lines between the bars are relaxed from being unobstructed to being obstructed by at most k other bars. Hartke et al. published Further Results on Bar k-Visibility Graphs [13], and in combination these two papers established that the maximum number of edges in a bar k-visibility graph on n vertices is (k + 1)(3n − 4k − 6). Dean et al. also proved that the thickness * of every bar 1-visibility graph is at most 4, and Chang et al. [4] proved that the thickness of a bar k-visibility graph is at most 3k + 3.
Others, such as Babbitt et al. [1], have studied k-visibility on other types of visibility representations. Here we define a rectangle k-visibility graph to be a graph that can be represented with vertices as disjoint axis-parallel rectangles, where there is an edge between two vertices if and only if there is an axis-parallel line of sight, obstructed by at most k other rectangles, between the corresponding rectangles. By the above, as edges corresponding to horizontal as well as vertical visibility lines form bar k-visibility graphs, the number of edges and the thickness in a rectangle k-visibility graph are at most 2(k + 1)(3n − 4k − 6) and 6k + 6, respectively. In particular, the respective thicknesses of rectangle 0-and 1-visibility graphs are at most 2 and 8.
Prior research has further generalized rectangle visibility graphs into 3 dimensions, where they are referred to as box visibility graphs [9]. Similar in spirit to the Euclidean dimension of a graph, which is the minimum number of dimensions for which it is a (strict) unit distance graph, here we consider a generalization of rectangle k-visibility graphs into higher dimensions and study the minimum dimension needed to represent various graphs with k-visibility for a fixed k. For example, as discussed above, the minimal embedding dimension (MED) of a planar graph given k = 0 is at most 2.
We study such MEDs on general graphs in Section 3. Among other things, we show that the MED of a nonempty graph G on n vertices is at most n 2 , that the MED of a disconnected graph G is the maximum of 2 and the MEDs of its connected components, and that MEDs are subadditive under the Cartesian product.
We then move on to specific graphs. We cover complete graphs in Section 4, where we establish that MEDs can be arbitrarily large and that the MED of the complete graph on m vertices K m is at most max 3, m−22( k/2 +1) 2(k+1) + 1 . In Section 5 on multipartite graphs we find that the MED of a complete r-partite graph is at most r + 1. Section 6 is devoted to hypercubes; we show that the MED of Q m is at most 2 3 m , and at most √ m for k = 0, m = 2. Finally, in Appendix A we compare selected results from this paper to results about other graph dimensions.

Terminology
We define a d-dimensional rectangle visibility graph (RVG s ) to be a graph where vertices can be represented as disjoint (closed hyper-)rectangles in d dimensions, and edges as all axis-parallel lines of sight between (i.e., unobstructed line segments connecting) these (hyper-)rectangles. We also define these variants: • An −visibility graph (RVG ) imposes a positive thickness to the line of sight between rectangles, such that the rectangles must overlap by a positive amount in all (d−1) dimensions orthogonal to the line of sight. In contrast, a strong visibility graph (RVG s ) allows visibility lines with zero thickness, i.e. zero overlap along orthogonal directions.
• A rectangle k-visibility graph (RkVG s , RkVG ) allows the line of sight to be obstructed by up to k other rectangles.
Unless explicitly stated, we use the term rectangle to mean d-dimensional hyper-rectangle. As a special case, a box is a 3-dimensional rectangle.
The minimal embedding dimension (MED) of a graph G is the smallest number of spatial dimensions d for the graph to be a specific one of the above. We denote by M s (G), µ s (G), M s k (G), µ s k (G), M (G), µ (G), M k (G), and µ k (G) the MEDs of G as a RVG s , URVG s , RkVG s , URkVG s , RVG , URVG , RkVG , and URkVG , respectively. Example 1 µ s 1 (C 4 ) = 2 is the smallest number of dimensions in which we can represent C 4 as a unit rectangle 1-visibility graph with strong visibility.
Additionally, we use the following conventions: • G will be a simple graph. (We do not consider the null graph on zero vertices.) • n := |V (G)| ≥ 1 is the number of vertices (i.e., size) of G.
• A graph is empty if it has no edges.
• The or s superscript may be omitted, in which case the strong and -visibility models can each be applied consistently.
Example 2 "G is an M (G)-dimensional RVG" is always true because G is an M s (G)-dimensional RVG s and G is an M (G)-dimensional RVG .
• All occurrences of µ M can be consistently replaced by either µ or M.
3 General Graphs

Existence of the MEDs
Here we will prove that the minimal embedding dimension is well-defined, i.e. that every graph has a minimal embedding dimension. To that end, we first show how to think of a representation of a d-dimensional (U)RkVG (for large k) in terms of its projections to the axes.
Definition 4 A graph G is an interval graph if there is a one-to-one correspondence between its vertices and a set of (closed) intervals, such that two intervals overlap if and only if there is an edge between the corresponding vertices in G.
A unit interval graph, more commonly known as an indifference graph, is an interval graph that can be represented with unit intervals. Figure 1: A graph G represented as a 2-dimensional RkVG, with projected intervals in each dimension corresponding to vertices in interval graphs G 1 and G 2 Lemma 1 A graph G with n vertices is a d-dimensional (U)RkVG s , where k ≥ n − 2, if and only if there exist d (unit) interval graphs G 1 , . . . , G d , each on the same vertex set as G, such that no edge is contained in all of G 1 , . . . , G d and two vertices u, v ∈ G are adjacent if and only if they are adjacent in all but exactly one of G 1 , . . . , G d .
Proof: We note that k ≥ n − 2 is the same as infinite visibility, as at most n − 2 rectangles can obstruct a visibility line between any two rectangles.
First we go from a d-dimensional (U)RkVG s G to corresponding (unit) interval graphs G 1 , . . . , G d .
Consider the projections of all rectangles onto each of the axes of R d . Let G i be the (unit) interval graph formed by the projection onto the i th axis. Two rectangles cannot overlap in all of these projections, lest they would themselves overlap. In other words, no edge can be in all of G 1 , . . . , G d .
Two rectangles can see each other via a visibility line in the direction of the i th axis (1 ≤ i ≤ d) if and only if their respective projections do not overlap on the i th axis, but overlap on all other axes j = i for (1 ≤ j ≤ d). In other words, two vertices G are adjacent if and only if they are adjacent in all but exactly one of G 1 , . . . , G d .
Then to construct a (U)RkVG s representation if we have a set of (unit) interval graphs G 1 , . . . , G d , we can simply take the arrangement of rectangles for which the (unit) rectangle projections onto the axes correspond to the (unit) interval representations of G 1 through G d .
With this in mind, we now construct a representation of any graph G as a (U)RkVG by specifying its projections.
Theorem 1 Every graph has a minimal embedding dimension as a (U)RkVG. Specifically, for a graph G on n vertices, µ M k (G) ≤ n.
Proof: Let G's vertex set be [n] = {1, 2, . . . , n}, and let Let G i be the (unit) interval graph formed by S i . Note that (a) In S i 's range, there is no interval strictly between two other intervals, By (a), no rectangle can block a visibility line between two others, and by (e), strong vs. -visibility doesn't matter, so G is also a d-dimensional URkVG.

Basic Properties
We now make the following observations about minimal embedding dimensions: Proof: Any representation of G as a URkVG in µ k (G) dimensions is also a valid representation of G as RkVG, thus M k (G) ≤ µ k (G). Proof: Given a representation of G as a (U)RkVG s , let δ i be the smallest nonzero difference between the i th coordinates of any two of its hyperrectangles. In each dimension i, expand the rectangles by a margin of δi 3 . No new strong-visibility lines have been created or destroyed, as the pairs of rectangles which overlapped have not changed in any dimension. Moreover, any two rectangles that previously had any overlap now have a positive overlap. Scaling the representation along the i th axis by a factor of 1/ 1 + 2 δi 3 so that the rectangles return to their original size, we now have a representation of G as a (U)RkVG , so µ M k (G) ≤ µ M s k (G), as desired.
Proof: The former implies the latter by definition.
The latter implies the former because we can take a representation of G in µ M k (G) dimensions, place it in d-dimensional space, and thicken it by 1 unit in the remaining d − µ M k (G) dimensions.

Lemma 5 For any nonempty graph
where ω(G) denotes the size of G's maximum clique.
Proof: No representation of G can have a visibility line between two vertices that passes through > ω(G) − 2 others, as that would form an > ω(G)-clique. Thus, all visibility lines in representations of G are (ω(G) − 2)-visibility lines, so any (a − 2)-visibility representation is a (b − 2)-visibility representation and vice versa, and additionally, any (a − 2)-visibility representation is an (ω(G) − 2)-visibility representation, as desired.
Because the chromatic number of G, χ(G), is at least ω(G), we get the following corollary: Corollary 5 Lemma 5 holds for χ(G) in place of ω(G). be an additional orthogonal 1-dimensional space. We will construct a rectangle visibility representation of G by constructing its projections onto these spaces.

MEDs as RkVGs
The projection of S m onto S m will be one of the arrangements in Figure 3, such that the visibility graph formed between the green rectangles is G m (all possible values of G m are covered in Figure 3). Figure 3: All possible projections of vertices v 4m−3 , . . . , v min(4m,n) into S m Let T m be the complimentary set S \ S m . We project every other vertex v i ∈ T m onto the same 2-dimensional subspace S m in such a way that each projection covers the central point O, and either overlaps or is adjacent to each vertex in v j ∈ S m . Some possible projections are illustrated as orange rectangles in Figure 4. We use the following rules: • If i < j, the projections of v i and v j in S m will not overlap; this counts as being disjoint in one dimension.
If not, this counts as being disjoint in a second dimension, thus precluding any axis-parallel visibility line between the corresponding rectangles.
We note that every vertex v i ∈ T m overlaps with point O in S m ; thus there are no more disjoint projections than those described here.
By construction, we now have a representation of G where all pairs of vertices (v i , v j ) are disjoint in one dimension if they are adjacent, and in two dimensions if they are non-adjacent. Moreover, there does not exist any third vertex v k that blocks visibility between v i and v j (in particular in S k 4 at O for k / ∈ S i , S j ), so no rectangle can block a visibility line. Thus by Lemma 1, this construction is a valid representation of G in n 2 dimensions.

Graph Composition
We now look at relationships between the MEDs and various graph compositions.

Disjoint Union
We find the minimal embedding dimensions of the disjoint union of two graphs: If D ≥ 2, the minimal embedding dimension of their disjoint union is µ M k (G 1 G 2 ) = D.
Proof: We will separately prove that µ representations exist for each of G 1 and G 2 in D dimensions. By placing both of these representations in the same D-space in such a way that that they are non-overlapping in at least 2 dimensions, i.e., diagonally, we ensure that there exists no visibility lines between any vertex in G 1 and any vertex in G 2 . Thus, this is a valid By removing all vertices of G 2 , we are not creating any new edges (unobstructing potential visibility lines) in G 1 as by definition no visibility line exists between two rectangles representing vertices in G 1 and G 2 , respectively. This means that µ Because graphs with MED ≤ 1 must be connected, it follows that Corollary 6 Given graphs G 1 and G 2 , the minimal embedding dimension of their disjoint union is By repeatedly applying Corollary 6, we obtain Corollary 7 Given two or more graphs G 1 , . . . , G m , From Theorem 1 and Corollary 7, we obtain: From Theorem 2 and Corollary 7, we obtain: Corollary 9 For a graph G on n vertices, where the largest connected component has m ≤ n vertices,

Cartesian Product
We now show that MEDs are subadditive under the Cartesian product of graphs.

Theorem 3
The minimal embedding dimension of the Cartesian product of two graphs G 1 and G 2 as (U)RkVGs is bounded by dimensions. Take representations of G 1 and G 2 in S 1 and S 2 , respectively. For any two rectangles r 1 and r 2 in these respective representations, let R r1,r2 be the rectangle in µ M k (G 1 ) + µ M k (G 2 ) dimensions of which r 1 and r 2 are projections. Note that there is an immediate bijection between {R r1,r2 | r 1 ∈ S 1 , r 2 ∈ S 2 } and vertices in G 1 G 2 , namely, for any R r1,r2 , take the vertex in G 1 G 2 formed by the vertices corresponding to r 1 and r 2 , respectively.
If R s1,s2 and R t1,t2 overlap then s 1 and t 1 overlap and s 2 and t 2 overlap, which is not possible unless (s 1 , s 2 ) = (t 1 , t 2 ). Take the (U)RkVG of these rectangles. Given two adjacent rectangles, assume WLOG that the visibility line between these two rectangles is parallel to S 2 . Then, in S 1 the projection of these two rectangles as well as any of the ≤ k rectangles that obstruct the visibility line overlap, and thus are the same projected rectangle. The projection of these two rectangles onto S 2 are adjacent, obstructed by the projections of the same ≤ k other rectangles. Conversely, if two rectangles in the projection onto S 1 are the same and in S 2 are adjacent or vice versa, the rectangles are adjacent.
Therefore, this is a valid representation of By repeatedly applying Theorem 3 on multiple graphs, we obtain: The minimal embedding dimension of the Cartesian product of multiple graphs G 1 , . . . , G m as a (U)RkVG is bounded by

Rooted Product
We turn our attention to the rooted product, introduced in [11] by Godsil and McKay.
Definition 11 Let G be a graph on n vertices, and let H be a sequence of n rooted graphs Definition 12 Given a representation of a graph as an RkVG, and an open half-space S with an axis-parallel (d − 1)-dimensional hyperplane boundary, the expansion of the representation by a distance L is formed by moving all the hyperrectangles' corners in S by a distance L orthogonally away from the hyperplane. The expansion of a representation of a graph is another representation of the same graph, as all relationships are preserved.

Definition 13
Given a representation of a graph G as an RkVG and a vertex v ∈ G with rectangle R, the inflation of the representation at v by distance L is formed by expanding it on each half-space not containing R with boundary containing a face of R. There is an i such that v i is in the induced copy of G but v i+1 is not, as otherwise the path would not leave the induced G. Let H ∈ H be the rooted graph corresponding to Then, to establish that is only connected to the rest of G(H) at one vertex, so G(H) can be expressed as H i (G i ) for some sequence of rooted graphs G i . Thus by the above,

Corona Product
We now look at the corona product, introduced by Frucht and Harary [10]. Theorem 5 The minimal embedding dimension of the corona product of two graphs G and H as a RkVG is bounded by Proof: By Remark 15 and Theorem 4, we have where H is as in Remark 15. We

Complete Graphs
We now construct arrangements of rectangles where every rectangle can see every other rectangle, thus giving the complete graph.

MEDs as URkVGs
Theorem 6 For all k ≥ 0, the minimal embedding dimension of the complete graph on m vertices, K m , as a (U)RkVG is bounded by Proof: Because we can remove a rectangle from any representation of K m+1 to get one of K m , we need only prove that for d ∈ N, We imitate the proof of Theorem 2. For the sake of avoiding repetition, we simply show how the green and orange (this time, unit) rectangles are arranged. In the rest of the proof, the differences are: • replace every occurrence of 4 with 5, • let S 1 , . . . , S n−3 5 be orthogonal three-dimensional spaces, • for n ≡ 1, 2, 3 (mod 5), let S n    If for i from 0 to 21 we place a corresponding rectangle with thickness δ ∈ (0, 1) in 3-dimensional space at height z = i above this plane, such that its projection to the plane is the rectangle labeled i, we obtain a 0-visibility representation of K 22 , as in Figure 21. Replace the i th rectangle with k 2 + 1 duplicates of it with thickness (height) δ k 2 +1 at heights i + j k 2 +1 for j ∈ 0, . . . , , k 2 + 1 . We now have a k-visibility representation of where all rectangles are visible from any of the four sides, as seen in Figure 22. (A visibility line between two rectangles with different projections passes through at most k/2 other rectangles with the same projection as each of the former and the latter rectangle). We then thicken this representation by one unit into each of the remaining (d − 3) dimensions.
Finally, in each dimension except the 3 rd (along whose axis we stack our 22( k/2 + 1) rectangles), we add k + 1 hyperrectangles in both directions from the center, at increasing distances and with increasingly large hyperfaces facing the center, such that each hyperrectangle has k-visibility to every other rectangle; i.e., such that the added rectangles in each dimension surround the entire representation up to that point. (See Figure 23 for an example). Along the i th axis, there are 2(k + 1) rectangles surrounding the center and the rectangles corresponding to prior axes, for a total of 2(d − 1)(k + 1) rectangles surrounding the center. As all rectangles, big and small, are k-visible to each other, we have a representation of K 2(d−1)(k+1)+22( k/2 +1) .
Because we can remove a rectangle from any representation of K m+1 to get one of K m , M k (K m ) is non-decreasing, so this gives a bound for the minimal embedding dimension of the complete graph:

Corollary 16
The minimal embedding dimension of the complete graph on m vertices K m as a RkVG is bounded by

Growth of the MEDs
Lemma 7 For some fixed k, let Then, K c 2d−2 cannot be represented in d dimensions with all visibility lines parallel.
To prove this lemma, we apply a technique used by Fekete et al. in Theorem 4 of [8].
Proof: We use induction on d.
Base case: d = 2 Assume for the sake of contradiction that such a representation exists, and assume WLOG that all visibility lines are vertical. Flatten all rectangles so that they are horizontal line segments. We now have a bar k-visibility representation, as defined in the introduction, of K c2 = K 4k+5 . Hartke et al. have shown, however, that this is impossible [13].
We assume for the sake of contradiction that such a representation exists, and assume WLOG that all visibility lines are parallel to the first axis. As all rectangles overlap in every other dimension, there is then a line parallel to the d th axis that passes through all c 2d−2 rectangles. Translate the coordinate system such that the origin lies on . Each rectangle has two faces orthogonal to the d th axis, one on each side of . Let F n and F n be the coordinates along the d th axis of the corresponding faces for the n th rectangle, where F n is negative and F n is positive.
Chung showed that every sequence of a 2 + 1 numbers has a subsequence of length a with one local maximum [5]. Thus there exists a subsequence r 1 , . . . , r c 2d−3 among our c 2d−2 rectangles such that the sequence −F r1 , . . . , −F rc 2d−3 , has one local maximum.
Likewise, among these c 2d−3 rectangles there is a sub-subsequence s 1 , . . . , s c 2d−4 such that the distance from to the second face of each orthogonal rectangle, F s1 , . . . , F sc 2d−4 , form another unimaximal progression.
Note that in the d th dimension, if rectangles s i and s k overlap for i < j < k, rectangle s j contains their overlap. Thus, the visibility lines between these rectangles are those of their projections into the first (d − 1) dimensions. By the inductive hypothesis, these c 2d−4 = c 2(d−1)−2 rectangles cannot form a complete graph, as desired.
Theorem 8 The range of µ M k (K m ) over m for fixed k is the set of nonnegative integers, Z ≥0 .
  (adopting the notation from Lemma 7), where R denotes the multicolor Ramsey number function. Assume for the sake of contradiction that K r is representable in d dimensions. Color each edge of K r by the axis parallel to its visibility line. As this is a coloring with d colors of the edges of K r , there is a monochromatic K c 2d−2 , contradicting Lemma 7. Thus, K r is not representable in d dimensions.
Thus, no finite number of dimensions can represent K m for all m ∈ N, so µ M k (K m ) takes on arbitrarily large values. Since µ M k (K 1 ) = 0, it then suffices to show that µ M k (K m+1 ) ≤ µ M k (K m ) + 1. Assume that we have a representation of K m in µ M k (K m ) dimensions. Add an extra dimension, thicken all the rectangles by 1 unit in this dimension, and replace one rectangle with two copies shifted by − 2 3 and 2 3 into the new dimension, respectively. Then, as all visibilities are maintained and the two copies can see each other, we have a representation of K m+1 in µ M k (K m ) + 1 dimensions, as desired.

Complete Multipartite Graphs
To construct complete multipartite graphs, we arrange the rectangles in a crosshatch, so to speak.
Theorem 9 For all k ≥ 0, the complete (d − 1)-partite graph (which is the empty graph for d = 2 and is K m1,...,m d−1 for d > 2) is a d-dimensional RkVG.
Proof: Take an m 1 × m 2 × · · · × m d−1 lattice in (d − 1)-space. For each of the (d − 1) axes, take all orthogonal (d − 2)-spaces that pass through lattice points. Add a small thickness to each of these spaces in their respective orthogonal dimensions. Cut all these spaces off to get axis-orthogonal hyperrectangles surrounding the lattice points.
Note that any pair of rectangles corresponding to spaces orthogonal to the same axis do not intersect, but rectangles corresponding to different axes do. This gives a bound for the minimal embedding dimension of the complete multipartite graph:

Corollary 17
The minimal embedding dimension of the complete r-partite graph as a RkVG is M k (K m1, ..., mr ) ≤ r + 1 for r > 1.

k-Visibility
Hypercubes are bipartite graphs, so by the proof of Corollary 5, 1-visibility lines need to be avoided.
Theorem 10 For all k ≥ 0, the minimal embedding dimension of the hypercube graph on 2 m vertices, Q m , as a (U)RkVG is bounded by Proof: Figure 26 shows a representation of Q 3 in 2 dimensions, so M k (Q 3 ) = µ k (Q 3 ) = 2. Since µ M k (Q 1 ) = 1, by Corollary 10 we get

Remark 18
In a 2-dimensional representation of a bipartite (U)RkVG G with n vertices for k > 0, by Corollary 5, we can treat k as infinite, so G is the union of the interval graphs G 1 and G 2 (see Definition 4) formed by the horizontal and vertical projections, respectively. Order the vertices of G 1 by the starting points of their intervals and map each edge to its larger vertex in the ordering. If two edges were mapped to the same vertex v, the starting point of v would be contained in all intervals corresponding to vertices in the two edges, forming a triangle, a contradiction. Thus, this map is injective, so as no edge maps to the smallest vertex, there are ≤ n − 1 edges in G 1 , similarly ≤ n − 1 in G 2 , and in total, ≤ 2(n − 1) in G.

0-Visibility
We now move on to 0-visibility, where as opposed to our previous construction, we do not have to worry about collinear rectangles.
Our 0-visibility representations of hypercubes will be arranged in grids, so to speak. For example, in the representation of Q 6 shown in Figure 27, the rectangles are organized in a 2 3 × 2 3 grid. In order to construct representations of hypercube graphs, we will first show how to construct the columns, then show how to combine them into the full grid.

Gray Code
Before we proceed, we need to introduce the reflective binary Gray code, which we will simply refer to as Gray code.

Definition 19
The Gray code is a reordering of the binary numeral system such that two successive values differ in only one bit (binary digit) [12].
Like standard numbering systems (e.g., binary), Gray code representations of a number are implicitly padded with an infinite number of 0's on the left, and any number i is represented with a finite number of 1's. The number zero is represented with only 0's.
Given the Gray code representation of a non-negative integer i−1, the representation of i is formed by flipping the j th digit from the right, where the rightmost digit is the 0 th digit and 2 j is the largest power of 2 that divides i.
In the following discussion, we will denote by G i,j digit #j of the Gray code representation of i, counting from the right such that G i,0 is the least significant digit.
Example 20 The (four digit) Gray code representation of numbers 0 through 15 are shown in Figure 28. We will make use of the following properties of Gray code: • It is the reflective binary code, where the representation of numbers 0, . . . , (2 k − 1) are repeated in reverse order for numbers 2 k , . . . , (2 k+1 − 1), except that the k th digit is 1 instead of 0 (with digit #0 being the rightmost). In other words, for all i < 2 n and j < n, • The map from nonnegative integers to their Gray code representations is a bijection.
• The parity of a number is the parity of the number of 1's in its Gray code representation. As a consequence, any two numbers whose Gray code representations differ in exactly one bit have different parities.

MEDs as URVGs
First we construct the unit rectangle columns: The d-dimensional RVG formed by cubes of side length 2 centered at points of the form This construction for d = 3 is shown in Figure 29. Figure 29: A 3-dimensional URVG representation of Q 3 (the x, y, and z axes are colored red, green, and blue, respectively, and the offset of the center of each cube from the x axis is indicated) Note that the constant (d + 2) does not effect the validity of this lemma as anything sufficiently large for the cubes to be disjoint (more than 2) would work. Choosing (d + 2) becomes useful later on.
Proof: We use induction on d.
This case trivially holds, as a two segments form a valid representation of Q 1 in 1-dimensional space.
Inductive step: d ⇒ d + 1 For the remainder of this proof, we refer to the j th coordinate of the center of a rectangle as its j th coordinate. Q d+1 is formed by two induced copies of Q d with an edge between every pair of corresponding vertices. In light of this, we split the representation into two sets: i < 2 d and i ≥ 2 d , and biject each of these sets to the representation of Q d .
For 0 ≤ i < 2 d , map the i th rectangle of the domain to the i th rectangle of the range, and for 2 d ≤ i < 2 d+1 , map the i th rectangle of the domain to the (2 d+1 − 1 − i) th rectangle of the range. We first show that these maps are graph isomorphisms. All visibility lines are parallel to the 1 st axis and the order of the first coordinates is preserved (or reversed). In addition, since G i,j = G 2 d+1 −1−i,j for j < d, this map preserves the 2 nd to (d − 1) th coordinates.
Thus after introducing the d th coordinate, within each group, all rectangles between two others with d th coordinate 0 have d th coordinate 0, and the same is true of d th coordinate 1. Thus, any obstructing rectangles in the image are obstructing rectangles in the domain, as desired.
It remains to show that the correct visibility lines are drawn between the two groups; more specifically, that two rectangles in different groups share an edge if and only if they are mapped to the same rectangle. Since the two rectangles mapping to rectangle i of the image are the only two rectangles intersecting the (parameterized) line it indeed holds that there is an edge between them. In addition, for any two rectangles in different groups whose images are not the same, say rectangles is between the two rectangles. In the former and latter cases rectangle N has the same coordinates (besides the first) as rectangles i 1 and i 0 , respectively, and thus blocks any possible visibility line between i 0 and i 1 .
Combining the above, we find that this is indeed a valid representation.
Now we arrange such columns into a grid: Proof: For any tuple (i 1 , i 2 , . . . , i d ) with 0 ≤ i j ≤ 2 d − 1 for all j ∈ {1, . . . , d}, take cubes of side length 2, where each cube is centered at the sum of the vectors . . .
There are 2 d 2 such tuples.
If we fix all but one of i 1 , i 2 , . . . , i d , the corresponding rectangles form a Q d by Lemma 8.
In addition, if WLOG i 1 > i 1 the smaller 1 st coordinate of the rectangle corresponding to (i 1 , i 2 , . . . , i d ) is more than the larger 1 st coordinate of the rectangle corresponding to (i 1 , i 2 , . . . , i d ), because  Thus, these are the only visibility lines, and no rectangles intersect.
If we shrink this construction by a factor of 2, all the cubes become unit rectangles. We see this construction applied to d = 0 through d = 3 in figures 30, 31, 32, and 33, respectively. By cutting the group of rectangles in half repeatedly, we then obtain the following corollary: Corollary 21 The minimal embedding dimension of the m-dimensional hypercube graph as a URVG is bounded by µ(Q m ) ≤ √ m .

MEDs as RVGs
We proceed similarly for normal rectangles, again by first constructing rectangle columns: The d-dimensional RVG formed by rectangles with opposite vertices a i and b i , where d ≥ 2, 0 ≤ i < 2 d+1 , and Again, the constant d + 4 does not effect the validity.
Proof: We use induction on d.
Base case: d = 2 The resulting representation is shown in Figure 34. Again, we show the validity of our construction for Q d+2 by splitting it into two groups and bijecting each group to the representation of Q d+1 . This time, our two groups will be the rectangles corresponding to 0 or 3 (mod 4), as well as those corresponding to 1 or 2 (mod 4). The maps will both send rectangle i of the domain to rectangle i 2 of the range.
All potential visibility lines are parallel to the 1 st axis, and are thus contained in some plane of the form x 3 = c, where c is a constant and x 3 is the coordinate along the 3 rd axis. Since the rectangles' projections along the 3 rd axis are all of the form [0, 4] or [1,5], the three distinct cross-sections where a visibility line could be are x 3 ∈ (0, 1), (1, 4), or x 3 ∈ (4, 5).
For x 3 ∈ (0, 1), the rectangles in the cross-section are those with G i,0 = 0, i.e., the first group. In addition, G i,j+1 = G i 2 ,j , so as the order of the first coordinates is preserved, the map on the first group sends two rectangles in the cross-section to two adjacent rectangles iff they are adjacent.
Similarly, for x 3 ∈ (4, 5), the rectangles in the cross section are those with G i,0 = 1, i.e., the second group, so similarly the visibility lines here are exactly the desired ones.
All rectangles appear in the cross-section x ∈ (1, 4), so there are no additional visibility lines between two rectangles in the same group. Taking a line that passes through all rectangles, we see that any rectangles i 0 = j 0 with i0 2 = j0 2 are adjacent. It thus suffices to show that the converse holds, i.e. that if rectangles i 0 in the first group and j 0 in the second group are adjacent, i0 2 = j0 2 . Assume for the sake of contradiction that i0 Again we arrange such columns into a grid: Proof: For any tuple (i 1 , i 2 , . . . , i d ) with 0 ≤ i j ≤ 2 d+1 − 1 for all j ∈ {1, 2, . . . , d}, take a rectangle with opposite corners at There are 2 d 2 +d such tuples. Note that subtracting the opposite coordinates of each rectangle gives that the side lengths are from 2 to 4.
If we fix all but one of i 1 , i 2 , . . . , i d , the corresponding rectangles form a Q d+1 by Lemma 8. In addition, if WLOG i 1 > i 1 the smaller 1 st coordinate of the rectangle corresponding to (i 1 , i 2 , . . . , i d ) is more than the larger 1 st coordinate of the rectangle corresponding to (i 1 , i 2 , . . . , i d ), because  (where x denotes x rounded to the nearest integer).

Remark 23
The minimal embedding dimension d of hypercube graphs Q m as RVGs include the following: • M (Q 0 ) = 0. A representation of Q 0 in 0 dimensions is shown in Figure 30.
• M (Q 1 ) = 1. A representation of Q 1 in 1 dimension is shown in Figure 31. Q 1 cannot be represented in 0 dimensions because visibility lines are 1-dimensional.  Figure 35. Q 5 , Q 4 , Q 3 and Q 2 can be obtained by repeatedly removing the right or top half of the rectangles in these configurations, thus ending up with 2 5 , 2 4 , 2 3 , and 2 2 rectangles in each respective representation.
None of these graphs can be represented in 1-space, where the only graphs that can be represented are paths. Thus, the minimal embedding dimensions of Q 2 through Q 6 are d = 2.
Q 11 , Q 10 , Q 9 and Q 8 can be obtained by repeatedly removing the top half of the boxes in these configurations, thus ending up with n = 2 11 , 2 10 , 2 9 , and 2 8 boxes in each respective representation.
Dean and Hutchinson found that a bipartite 2-dimensional RVG on n ≥ 4 vertices has at most 4n − 12 edges [7]. Given m ≥ 8, the number of edges in Q m is m 2 · 2 m ≥ 4 · 2 m = 4n. Since Q m is bipartite, and 4n > 4n − 12, it follows that Q m≥8 is not representable in 2 dimensions. Thus, the minimal embedding dimensions of Q 8 through Q 12 are all d = 3.

Open Questions
Some questions not answered in this paper but possibly worth exploring in future works include: 7. For fixed k, can the MED as a (U)RkVG of the composition (under e.g., the tensor product or the strong product) of two graphs with bounded MEDs be unbounded?

A MEDs vs. Other Dimensions
Here, we compare MEDs of graphs as (U)RVGs with other dimensions of graphs, in particular: Definition 24 The Euclidean dimension Edim(G) of a graph G is the minimum number of dimensions required to represent its vertices as points such that two points have distance 1 if and only if they share an edge.
Definition 25 The metric dimension β(G) of a connected graph G is the minimum number of vertices that need to be selected in G for every vertex to be uniquely identified by its distance to each selected vertex.
In the table shown in Figure 38, G 1 , G 2 , and G are nonempty graphs.