aa r X i v : . [ m a t h . C O ] S e p Unit Hypercube Visibility Numbers of Trees
Eric Peterson ∗ and Paul S. Wenger † September 28, 2018
Abstract
A visibility representation of a graph G is an assignment of the vertices of G togeometric objects such that vertices are adjacent if and only if their correspondingobjects are “visible” each other, that is, there is an uninterrupted channel, usuallyaxis-aligned, between them. Depending on the objects and definition of visibility used,not all graphs are visibility graphs. In such situations, one may be able to obtain avisibility representation of a graph G by allowing vertices to be assigned to more thanone object. The visibility number of a graph G is the minimum t such that G has arepresentation in which each vertex is assigned to at most t objects.In this paper, we explore visibility numbers of trees when the vertices are assignedto unit hypercubes in R n . We use two different models of visibility: when lines of sightcan be parallel to any standard basis vector of R n , and when lines of sight are onlyparallel to the n th standard basis vector in R n . We establish relationships betweenthese visibility models and their connection to trees with certain cubicity values. Keywords: 05C62, visibility, cubicity
Broadly speaking, a visibility representation of a graph G is an assignment of the vertices of G to geometric objects embedded in an ambient space so that two vertices are adjacent if andonly if there is a line of sight between their objects that intersects none of the other objects.Visibility representations have been studied with a wide variety of geometric objects includingbars [12, 19, 22], semi-bars [6], rectangles [3, 8, 9], points [15], and circular arcs [16, 17].Historically, the study of visibility representations was motivated by VLSI design. Re-flecting these roots in circuit design, it may be inappropriate to have vertices assigned toobjects whose sizes can differ by arbitrary amounts, since the components of electroniccircuits are roughly uniform in size. A natural restriction is to require that all vertices are ∗ Dept. of Mathematics, Univ. of Rhode Island, Kingston, RI; [email protected] . † School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY; [email protected] v v v v v v v v v v v Figure 1: A unit bar visibility representation and a unit rectangle visibility representationof a graph.assigned to objects of the same size. Such representations have been studied with bars [7, 11]and rectangles [10].In this paper we study visibility representations in which vertices are assigned to unithypercubes in R n . Throughout we will use the standard basis vectors e , . . . , e n as a basisfor R n . We consider two different versions of visibility, which are inspired by unit barvisibility graphs and unit rectangle visibility graphs. In a unit bar visibility representation of a graph, vertices are assigned to disjoint horizontal bars of length 1 in the plane, andtwo bars see each other if there is a vertical channel of positive width joining them thatintersects no other bar (see Figure 1). Unit bar visibility graphs were first studied by Deanand Veytsel [11]. In a unit rectangle visibility representation of a graph, vertices are assignedto disjoint axis-aligned unit rectangles in the plane, and two rectangles see each other ifthere is a vertical or horizontal channel of positive width joining them that intersects noother rectangle (see Figure 1). Unit rectangle visibility graphs were first studied by Dean,Ellis-Monaghan, Hamilton, and Pangborne [10].There is an important distinction between the models of unit bar and unit rectanglevisibility graphs: in a unit bar visibility graph, the lines of sight are orthogonal to theaffine spaces defined by the objects; in a unit rectangle visibility graph, the lines of sight areorthogonal to faces of the rectangles, but live in the same ambient space as the objects. Thereis also a clear connection between unit bar and unit rectangle visibility representations: thehorizontal and vertical lines of sight in a unit rectangle visibility representation corresponddirectly to unit bar visibility representations.To elucidate this distinction, we define two versions of visibility for unit hypercube visibil-ity graphs. Throughout the paper, we use n -cube to mean an n -dimensional unit hypercube.A graph G has an n -cube visibility representation if the vertices of G can be assigned todisjoint axis-aligned unit n -cubes in R n such that two vertices are adjacent if and only ifthere is an uninterrupted axis-aligned cylindrical channel of positive diameter between their2igure 2: A unit 2-cube orthogonal visibility representation of K , and a unit 2-cube visi-bility representation of K , .respective n -cubes. A graph with an n -cube visibility representation is an n -cube visibilitygraph . In this setting, a unit rectangle visibility graph is a 2-cube visibility graph. A graph G has an n -cube orthogonal visibility representation if the vertices of G can be assigned todisjoint unit n -cubes in R n +1 such that the cubes are aligned with the first n axes, and twovertices are adjacent if and only if there is an uninterrupted cylindrical channel of positivediameter that is parallel to e n +1 between their respective n -cubes. A graph with an n -cubeorthogonal visibility representation is an n -cube orthogonal visibility graph . In this setting,a unit bar visibility graph is a 1-cube orthogonal visibility graph. See Figure 2 for examplesof a 2-cube visibility representation and a 2-cube orthogonal visibility representation.The family of graphs that have a specified type of visibility representation is generallyquite limited. For instance, when objects are embedded in the plane and the directions oflines of sight are limited, there usually is a bound on the thickness of such a graph. Toaddress such limitations, one can assign vertices to more than one object in a visibilityrepresentation. This approach was first taken by Chang, Hutchinson, Jacobson, Lehel, andWest [5] in the context of bar visibility representations. The bar visibility number of a graph G is the minimum t such that G has a bar orthogonal visibility representation in whicheach vertex is assigned to at most t bars. Bar visibility numbers were further studied byAxenovich, Beveridge, Hutchinson, and West [1] for directed graphs (edges are orientedtowards whichever bar has a larger y -coordinate), and also by Gaub, Rose, and the secondauthor [13] when all bars have unit length. 3 v v v v v v xv v Figure 3: A 2-cube visibility representation of K , using at most two cubes per vertex. Therepresentation has two components.In this paper, we study visibility numbers of trees when vertices are assigned to sets ofunit hypercubes. The n -cube visibility number of a graph G , denoted h ( n ) ( G ) is the minimum t such that there is a n -cube visibility representation of G in which each vertex is assigned toat most t unit n -cubes in R n . The n -cube orthogonal visibility number of a graph G , denoted h ( n ) ⊥ ( G ) is the minimum t such that there is an n -cube orthogonal visibility representationof G in which each vertex is assigned to at most t n -cubes in R n +1 (recall that these cubesare orthogonal to e n +1 ). See Figures 3 and 4 for examples.In all n -cube visibility representations and n -cube orthogonal visibility representationswe will add the additional requirement that no two cubes that are assigned to the samevertex can see each other. Such lines of sight would allow us to use multiple n -cubes to actlike an n -dimensional box with non-unit dimensions.It is clear that every tree has bar visibility number 1 (here the bars can have differentlength). In [13], Gaub et al. presented a linear-time algorithm that determines the 1-cubeorthogonal visibility number of any tree. In this paper, we study the relation between n -cube visibility numbers and ( n − v v v xv v v v v xv Figure 4: A 2-cube orthogonal visibility representation of K , using at most two cubes pervertex. The representation has two components.their connections to related arboricity parameters. We also characterize trees with n -cubeorthogonal visibility representations in terms of trees that have representations as intersectiongraphs of n -cubes in R n . A tree is an n -cube (orthogonal) visibility tree if it is an n -cube (orthogonal) visibility graph.A graph is an n -cube (orthogonal) visibility forest if it is the disjoint union of n -cube (or-thogonal) visibility trees. Let Υ h ( n ) ( G ) denote the minimum k such that G is the union of k unit n -cube visibility forests. Let Υ h ( n ) ⊥ ( G ) denote the minimum k such that G is the unionof k unit n -cube orthogonal visibility forests. We call Υ h ( n ) ( G ) and Υ h ( n ) ⊥ ( G ) the n -cubearboricity and the n -cube orthogonal arboricity of G , respectively.We begin by showing that the n -cube (orthogonal) visibility number of a tree T is equalto the n -cube (orthogonal) arboricity of T . Since the same proof works for both standard andorthogonal visibility models, we use h ( n ) ∗ ( T ) and Υ h ( n ) ∗ ( T ) to indicate the visibility numberand arboricity, where the visibility may be standard or orthogonal.Let R be an n -cube (orthogonal) visibility representation of a graph G . A component of R is a maximal set of n -cubes that are connected by the visibility relation. An importantaspect of the visibility models that we study is that it is possible for a visibility representationto have multiple components. This is not true in some visibility models, notably the pointvisibility graphs studied in [15]. Let G and G ′ be graphs with visibility representations R and R ′ respectively ( R and R ′ are assumed to be representations of the same type). We canobtain a visibility representation of G ∪ G ′ by adding the cubes of R ′ to R , and translatingthe cubes of R ′ all by the same vector so that no cube from R sees a cube of R ′ ; we call5he resulting representation a disjoint union of R and R ′ . The representations in Figures 3and 4 can both be thought of as the disjoint union of two components.We require the following technical lemma, a version of which appeared for n = 1 asLemma 1 in [13]. Since the proof is essentially identical to the proof in [13], we omit it here. Lemma 1. If G is a unit n -cube orthogonal visibility graph, then there is a unit n -cubeorthogonal visibility representation of G in which all n -cubes have distinct coordinates in thedirection of e n +1 . For ease of discussion, we will refer to the ( n + 1)st coordinate of an n -cube in an n -cubeorthogonal visibility representation as its height . Theorem 2. If T is a tree, then Υ h ( n ) ∗ ( T ) = h ( n ) ∗ ( T ) .Proof. If Υ h ( n ) ∗ ( T ) = k , then there exists a decomposition of T into k n -cube (orthogonal)visibility forests. A disjoint union of the n -cube (orthogonal) visibility representations ofeach of these forests is an n -cube (orthogonal) visibility representation of T in which eachvertex is assigned to at most k cubes, so h ( n ) ∗ ( T ) ≤ Υ h ( n ) ∗ ( T ).Let h ( n ) ∗ ( T ) = t and let R be an n -cube (orthogonal) visibility representation of T in which each vertex is assigned to at most t cubes. If R is an n -cube orthogonal visibilityrepresentation, then assume by Lemma 1 that all n -cubes have distinct heights. Furthermore,assume that R is chosen so that it contains the minimum number of pairs of n -cubes thatcorrespond to the same vertex and lie in the same component of R .Assume that there exists a v ∈ V ( T ) such that two n -cubes corresponding to v lie inthe same component of R ; call the component R . For each edge xy represented as a line ofsight in R , choose a line of sight between cubes for x and y in R of minimum Euclideanlength, breaking ties arbitrarily. The chosen lines of sight create a spanning forest F of the n -cubes in R . Because T is a tree, there is at most one cube assigned to each vertex in eachcomponent of F . Let R ′ be a representation obtained by taking the disjoint union of R − R and each of the visibility representations of the components of F . Observe that R ′ retainsall edges represented in R and contains fewer pairs of hypercubes that are assigned to thesame vertex and lie in the same component.It remains to show that R ′ does not have any lines of sight that correspond to edges thatare not in T . Suppose that the n -cubes B ( a ) and B ( b ) correspond to vertices a and b thatare not adjacent in T , but B ( a ) and B ( b ) see each other in R ′ ; clearly B ( a ) and B ( b ) lie in R in the representation R . Thus, there is a channel of visibility between B ( a ) and B ( b ) in R ′ that does not exist in R . Therefore, there is a collection of n -cubes in R that blocks6 ( a ) B ( b ) B ( c ) B ( c ) B ( c ) B ( c k )... Figure 5: A 2-cube visibility representation with cubes B ( c ),..., B ( c k ) blocking B ( a ) from B ( b ). The blocking cubes are in a different component from B ( a ) and B ( b ) when R ispartitioned.the channel of visibility between B ( a ) and B ( b ), and these blocking cubes are in a differentcomponent of the spanning forest of R . Label the blocking cubes B ( c ) , B ( c ) , ..., B ( c k )where B ( a ) sees B ( c ), B ( c i ) sees B ( c i +1 ) for i ∈ [ k − B ( c k ) sees B ( b ) (see Figure 5).Note that ac . . . c k b is a walk in T between a and b ; call the walk W . Let the length ofthe channel between B ( a ) and B ( b ) be d . The sum of the Euclidean lengths of the lines ofsight corresponding to edges in W is d − k if R is an n -cube visibility representation and d if R is an n -cube orthogonal visibility representation.Let P be the unique path in T with endpoints a and b , and let P have length m + 1.Thus, m ≤ k . Since P is contained in the component of B ( a ) and B ( b ), the sum of theEuclidean lengths of the lines of sight corresponding to edges in P is at least d − m if R isan n -cube visibility representation and is at least d if R is an n -cube orthogonal visibilityrepresentation. Note that the m + 1 edges in P are also represented in W . If m < k , thenthere is an edge in P with a line of sight in W that is shorter than its line of sight in P ,contradicting the assumption that the shortest line of sight for each edge has been chosen.Therefore, m = k , and W = P . If W does not contain a line of sight that is shorter thanthe corresponding line of sight in P , then the length of each line of sight in W is equal tothe length of the corresponding line of sight in P . Therefore, in R there are parallel lines ofsight of the same length that join B ( a ) to two n -cubes corresponding to the same vertex in T .If R is an n -cube visibility representation, this contradicts the assumption that no n -cubescorresponding to the same vertex can see each other. If R is an n -cube orthogonal visibilityrepresentation, this contradicts the assumption that all cubes in R have distinct heights. Weconclude that there are no lines of sight in R ′ that correspond to edges that are not in T .Thus, R ′ is a representation of T with fewer pairs of n -cubes that are assigned to the samevertex and lie in the same component, contradicting the minimality of R . Therefore, thereis a representation of T that has the same number of n -cubes as R and is a disjoint union ofrepresentations of trees, so Υ h ( n ) ∗ ( T ) ≤ h ( n ) ∗ ( T ). Thus, h ( n ) ∗ ( T ) = Υ h ( n ) ∗ ( T ).7e now characterize n -cube visibility trees in terms of ( n − n = 2. The proof of Theorem 4.5 from [10] uses the structuralcharacterization of 1-cube orthogonal visibility trees due to Dean and Veytsel [11] to build2-cube visibility representations of trees. Our proof does not require a characterization ofunit ( n − n − n − n − Theorem 3.
A tree T is an n -cube visibility graph if and only if it is the union of n ( n − -cube orthogonal visibility forests.Proof. Let R be an n -cube visibility representation of T . By Theorem 2, we may assumethat all components of R are trees. For each i ∈ [ n ], the lines of sight in R that are parallelto e i correspond to a unit ( n − T . The union of these n forests is T .Now assume that T is a t -vertex tree that is the union of n ( n − F , . . . , F n (we may assume that these forests are spanning, since we canadd isolated cubes to their representations). Note that distinct components of F j and F j ′ share at most one vertex. Label the vertices of T using a breadth-first search starting at anarbitrary vertex v . For each j , label each subtree of T that is a component of F j as T i,j where i is the lowest index of a vertex in the tree.We will iteratively add cubes for the vertices in the trees T i,j in lexicographic order untilwe obtain a representation of T . When T i,j is processed, v i has already been assigned a cube.Assume that the cube B is added at ( x , . . . , x n ) when T i,j is processed. We will ensure thatthe following conditions are satisfied:(1) When B is added, it will only see the cubes of its neighbors in T i,j , and it will see thosecubes in the direction of e j . 82) When B is added, for each j ′ = j , there is no cube whose location lies in the set { ( x ± t, . . . , x j ′ − ± t, z, x j ′ +1 ± t, . . . , x n ± t ) | z ∈ R } .Begin by placing the cube for v at the origin. The component T , of F that contains v has an ( n − n − n -cubes for thevertices in T , so that (i) all lines of sight between them are parallel to e , (ii) they form arepresentation of T , , and (iii) their first coordinates all differ by at least 4 t . Condition 1clearly holds for each cube, and condition 2 holds since the j th coordinates of the cubes alldiffer by at least 4 t .To process T i,j , observe that v i has been assigned an n -cube. Let B i be the cube assignedto v i , and assume that B i was added to the representation when processing a component of F j ′ . By Lemma 1, T i,j has a unit ( n − R so that (a) the height of thecube in R that is assigned to v i is the j th coordinate of B i , (b) every other ( n − j th coordinate ofall existing n -cubes in the partial representation of T by at least 4 t , and (c) the heights ofall the ( n − t . Align such a representationso that the visibilities are parallel to e j , and for each v ∈ V ( T i,j ) − v i add an n -cube in thelocation of the corresponding ( n − R .Let B be a cube that is added when T i,j is processed. Since T i,j is connected and has atmost t vertices, each coordinate of B except the j th differs from the corresponding coordinateof B i by at most t . Since all cubes that are not in T i,j have locations that differ from thelocation of B i by at least 4 t in a direction that is not parallel to e j , it follows that B canonly see the cubes of vertices in T i,j . Since the j th coordinate of B also differs from the j thcoordinate of all other cubes by at least 4 t , conditions (1) and (2) hold for B .It follows that by processing all T i,j , we obtain an n -cube visibility representation of T . Theorem 3 allows us to relate Υ h ( n ) ( T ) to Υ h ( n − ⊥ ( T ) for a tree T . Theorem 4. If T is a tree, then Υ h ( n ) ( T ) = (cid:24) Υ h ( n − ⊥ ( T ) n (cid:25) . Proof.
Suppose that Υ h ( n ) ( T ) = k , and decompose T into k unit n -cube visibility forests.Each unit n -cube visibility forest is the union of n unit ( n − l Υ h ( n − ⊥ ( T ) n m ≤ Υ h ( n ) ( T ).9ow suppose that Υ h ( n − ⊥ ( T ) = ℓ . It follows that T can be decomposed into ( n − ℓ of those trees.Furthermore, by adding multiple copies of the 1-vertex tree at each vertex, we can obtain amultiset of ( n − T such that (i) everyvertex is contained in exactly ℓ members of the multiset, and (ii) the union of the trees inthe multiset is T . Construct an auxiliary graph G in which each tree of the multiset is avertex, and two vertices are adjacent if and only if they share a vertex in T . Since G is theintersection graph of subtrees of a tree, it follows that G is a chordal graph [4, 14, 20, 21].Hence G is perfect and χ ( G ) = ω ( G ), where χ ( G ) is the chromatic number of G and ω ( G )is the size of the largest clique in G . Furthermore, since the intersection of any two treesin the decomposition is at most a single vertex, ω ( G ) = ℓ . Therefore, there is a coloring ofthe trees in the decomposition using ℓ colors so that no two trees of the same color sharea vertex. By Theorem 3, the union of n ( n − n -cube visibility tree, so the union of n color classes of the trees in the multiset is an n -cubevisibility tree. Thus, T can be decomposed into ⌈ ℓ/n ⌉ n -cube visibility trees. Therefore, l Υ h ( n − ⊥ ( T ) n m ≥ Υ h ( n ) ( T ).In [13], Gaub et al. developed a fast algorithm to determine h (1) ⊥ ( T ) for a tree T andobtain a 1-cube orthogonal visibility representation of T . Theorem 5 (Gaub et al. [13]) . Let T be a tree. If ∆( T ) , then h (1) ⊥ = ⌈ ∆( T ) / ⌉ . If ∆( T ) ≡ , then h (1) ⊥ = ⌈ ∆( T ) / ⌉ or h (1) ⊥ = ⌈ (∆( T ) + 1) / ⌉ , andthere is a linear time algorithm that determines the correct value. Using this algorithm and Theorem 4, we can determine the 2-cube visibility number of atree.
Theorem 6.
Let T be a tree. If ∆( T ) , then h (2) ( T ) = (cid:24) ∆( T )6 (cid:25) . If ∆( T ) ≡ , then h (2) ( T ) = (cid:24) ∆( T )6 (cid:25) or h (2) ( T ) = (cid:24) ∆( T ) + 16 (cid:25) , and there is a linear time algorithm to determine the exact value. roof. If ∆( T ) l ∆( T )3 m ≤ h (1) ⊥ ( T ) ≤ l ∆( T )+13 m , and by Lemma 4, h (2) ( T ) = l h (1) ⊥ ( T )2 m = l ∆( T )6 m . If ∆( T ) ≡ h (1) ⊥ ( T ) = l ∆( T )3 m or h (1) ⊥ ( T ) = l ∆( T )+13 m , and there is an algorithm to determine the the correct value in linear time. There-fore, h (2) ( T ) = l ∆( T )6 m or h (2) ( T ) = l ∆( T )+16 m and there is an algorithm to determine thecorrect value.The algorithm UNIT BAR TREE from [13] that determines the value of h (1) ⊥ ( T ) fora tree depends on the Dean-Veytsel characterization of 1-cube orthogonal visibility treesfrom [11]. Thus, similar results in the vein of Theorem 6 for higher dimensions would requireand understanding the structure of n -cube orthogonal visibility trees for n ≥ intersection representation of a graph G is an assignment of the vertices of G to setssuch that two vertices are adjacent if and only if their sets have nonempty intersection. Agraph has cubicity n if it has an intersection representation where each vertex is assigned toan axis-aligned closed unit n -cube in R n , but it does not have an intersection representationwhere each vertex is assigned to an axis-aligned closed unit ( n − R n − . Note thatif a graph has cubicity n , then it has a representation as an intersection graph of a set ofunit m -cubes for all m ≥ n . We show that the structure of unit n -cube orthogonal visibilitytrees is closely related to the structure of trees with cubicity at most n .First we address a slight discrepancy: in our visibility representations we require lines ofsight to be cylindrical channels with positive diameter, and in representations as intersectiongraphs of cubes we may edges represented by intersections that do contain any ǫ -ball (whichwould correspond to the diameter of the channel in the visibility graph). Lemma 7. If T is tree with cubicity n , then there is a representation of T as the intersec-tion graph of n -cubes and an ǫ > such that such that all nonempty intersections of therepresentation contain an ǫ -ball.Proof. Let R be an intersection representation of T with the minimum number of intersec-tions that do not contain an ǫ -ball for all ǫ >
0. If there are no such intersections, then theresult holds. Assume that u and v are two adjacent vertices whose cubes intersect in a setthat contains no ǫ -ball for every ǫ >
0. Let T u be the component of T − uv that contains u , and let T v be the component that contains v . There exists δ > u ′ , v ′ ) where u ′ ∈ T u and v ′ ∈ T v and ( u ′ , v ′ ) = ( u, v ) the cubesof u ′ and v ′ have locations that differ by at least 1 + δ in every coordinate. Let w be thevector δ (sgn( v − u ) , . . . , sgn( v n − u n )) where sgn denotes the sign function; translating the11ube of u by w will move it towards the cube of v by δ in every direction. For each vertex u ′ ∈ V ( T u ), translate the cube of u ′ by w . The only intersection of cubes affected by thistranslation is the intersection of the cubes of u and v , which now contains a δ/ R .Let T be a tree and let v ∈ T . A path expansion of length k at v is the process of replacing v by a path v , v , . . . , v k and partitioning the set of neighbors of v so that each neighbor of v is now adjacent to exactly one of v or v k . Theorem 8.
A tree T is a unit n -cube orthogonal visibility graph if and only if it can beobtained from a tree T ′ with cubicity at most n by performing a path expansion at each vertexof T ′ (perhaps some with length ).Proof. Let T be obtained from the tree T ′ with cubicity at most n by performing a pathexpansion at each vertex of T ′ . Consider an n -cube intersection representation of T ′ in whichall nonempty intersections contain an ǫ -ball for some ǫ >
0, which is guaranteed by Lemma 7.Because T ′ contains no cycles, it follows that each point in R n is contained in the n -cubesof at most two vertices in T ′ . Place the representation of T ′ into R n +1 so that the cubesoccupy the subspace spanned by { e , . . . , e n } .Fix a vertex v in T ′ . Starting from v we 2-color the edges of T ′ using a breadth-firstsearch in the following fashion. When performing the path-expansion at v to obtain T , theneighbors of v in T ′ are partitioned into two sets N ( v ) and N ( v ). Color the edges joining v and vertices in N ( v ) blue, and color the edges joining v and vertices in N ( v ) red. Whenprocessing the edges at a vertex u = v , exactly one edge uu ′ is already colored, where u ′ is the predecessor of u in the breadth first search. Suppose that u ′ ∈ N i ( u ) when the pathexpansion at u is performed. Color all edges joining u to vertices in N i ( u ) the same color as uu ′ . Color all other edges at u with the other color.Now place the n -cube for v in the subspace of R n +1 spanned by { e , . . . , e n } (so that thevalue of the ( n + 1)st coordinate is 0). For u ∈ V ( T ), let b vu denote the number of blueedges on the path from v to u , and let r vu denote the number of red edges on the path from v to u . For each u in v , let the ( n + 1)st coordinate of the corresponding n -cube in R n +1 begiven by b vu − r vu . This is a unit n -cube orthogonal visibility representation of T ′ .Let v be a vertex in T ′ where a path expansion is performed that produces a path oflength k . To construct a unit n -cube orthogonal visibility representation of T , place a stackof k evenly spaced unit n -cubes at heights between b vu − r vu − and b vu − r vu + .12ow suppose that T has an orthogonal n -cube representation. Project the cubes ontothe subspace of R n +1 spanned by { e , . . . , e n } . The result is an intersection representationusing unit n -cubes of a tree T ′ with cubicity at most n . The cubes that project onto thesame cube form a path in T , and this process can be reversed using path expansions.Trees with cubicity 1 are clearly paths, so the characterization of unit bar visibility graphsis a corollary to Theorem 8. Theorem 9. [Dean and Veytsel [11]] A tree is a unit bar visibility graph if and only if it isa subdivided caterpillar with maximum degree . At the moment, results similar to Thoerem 9 appear out of reach since the complexity ofdetermining even if a tree has cubicity 2 is still unknown [2]. However, we are able to givevery strong bounds on both the n -cube visibility numbers and n -cube orthogonal visibilitynumbers of trees that are equal in many cases. Theorem 10. If T is a tree, then (cid:24) ∆( T )2 n + 1 (cid:25) ≤ h ( n ) ⊥ ( T ) ≤ (cid:24) ∆( T ) + 12 n + 1 (cid:25) and (cid:24) ∆( T ) n (2 n − + 1) (cid:25) ≤ h ( n ) ( T ) ≤ (cid:24) ∆( T ) + 1 n (2 n − + 1) (cid:25) . Proof.
First note that if a tree T has cubicity n , then ∆( T ) ≤ n . Therefore, by Theorem 8,if T is an n -cube orthogonal visibility graph, then ∆( T ) ≤ n + 1 (obtained by performinga path expansion in which the partition of the neighbors of v has only one nonempty set).Therefore, a decomposition of a tree T into n -cube orthogonal visibility forests requires atleast l ∆( T )2 n +1 m forests.By Theorem 3, if T is a n -cube visibility graph, then ∆( T ) ≤ n (2 n − + 1). Therefore, adecomposition of a tree T into n -cube orthogonal visibility forests requires at least l ∆( T ) n (2 n − +1) m forests.Also, note that K , n has cubicity n . It then follows from Theorem 8 that K , n +1 isan n -cube orthogonal visibility graph, and from Theorem 3 that K ,n (2 n − +1) is an n -cubevisibility graph. A tree T can be greedily decomposed into l ∆( T )+1 k m forests with maximumdegree k in which every component is a star.13 Conclusion
We conclude with some open problems and directions for further research.
Question 1.
For n ≥
2, is there a simple characterization of trees with n -cube orthogonalvisibility representations?Theorem 8 suggests the following more fundamental question, which is not being posedfor the first time here (see [2] and references therein). Question 2.
For n ≥
2, is there a simple characterization of graphs with cubicity n ?Of course one can also consider visibility numbers of graphs that are not trees. In hismaster’s thesis [18], the first author studied rectangle and unit rectangle visibility numbers ofa variety of graphs including complete graphs and complete bipartite graphs. The followingis perhaps the most natural starting point for a systematic study of visibility numbers inhigher dimensions. Question 3.
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