More directions in visibility graphs

In this paper we introduce unit bar k-visibility graphs, which are bar kvisibility graphs in which every bar has unit length. We show that almost all families of unit bar k-visibility graphs and unit bar k-visibility graphs are incomparable under set inclusion. In addition, we establish the largest complete graph that is a unit bar k-visibility graph. As well, we present a family of hyperbox visibility graphs that provide edge maximal rectangle visibility graphs in every possible standard 2-dimensional cross-section. We end with a list of open questions. 1 Background and Definitions Classes of visibility graphs have applications in VLSI design and graph layout [6]. Let R be a set of horizontal closed line segments, or bars, in the plane, at distinct heights. We say that a graph G is a bar visibility graph, and R a bar visibility representation of G, if there exists a one-to-one correspondence between vertices of G and bars in R, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. Formally, two vertices x and y in G are adjacent if and only if, for their corresponding bars X and Y in R, there exists a vertical line segment , called a line of sight, whose endpoints are contained in X and Y , respectively, and which does not intersect any other bar in S. A bar k-visibility graph is a graph with a bar visibility representation in which a line of sight between bars X and Y intersects 56 ELLEN GETHNER AND JOSHUA D. LAISON at most k additional bars [4]. A unit bar k-visibility graph is a graph which has a bar k-visibility representation in which every bar has unit length. Bar 0-visibility graphs are bar visibility graphs. Similarly, unit bar 0-visibility graphs are unit bar visibility graphs. The characterization of unit bar visibility graphs was begun in [5]. On the other hand, bar k-visibility graphs are interval graphs for large enough values of k. So bar k-visibility graphs can be thought of as existing between bar visibility graphs and interval graphs. A proper interval graph is a graph that has an interval representation in which no interval is properly contained in another [1]. Analogously, a proper bar kvisibility graph is a graph that has a bar k-visibility representation in which no bar contains another bar when considered as intervals. We omit the proof of the following proposition since it is a straightforward modification of the technique given in [1]. Proposition 1 A graph is a unit bar k-visibility graph if and only if it is a proper bar k-visibility graph. A rectangle visibility graph (RVG) is a graph G whose vertices can be represented in the plane by a set R of closed disjoint rectangles whose sides are parallel to the xand y-axes. Two vertices u and v in G are adjacent if and only if their corresponding rectangles rv and ru in R have a line of sight between them, parallel to one of the axes, that intersects no other rectangle in R. The analogy in d dimensions is a d-box visibility graph in which each vertex is represented by a d-dimensional hyperbox whose sides are parallel to the standard axes in R; two vertices are adjacent if and only if there is an unobstructed line of sight between them, parallel to one of the standard axes. We define d-box visibility graphs formally in Section 4. There are two different standard definitions of visibility graphs, using lines of sight and non-degenerate rectangles of sight. In general more edges may be blocked using rectangles of sight, and thus more graphs can be represented in this way. In [4, 3] it is shown that there exists a representation of an edge-maximal visibility graph G using lines of sight if and only if there exists a representation of G using bands of sight in two dimensions; the proof generalizes to d dimensions as well. That is, there exists an edge-maximal visibility graph G using lines of sight if and only if there exists a representation of G using non-degenerate d-dimensional bands of sight. Our present interest is in edge-maximal graphs, and thus we use lines of sight. 2 Unit Bar k-Visibility Graphs Let G be a unit bar k-visibility graph and R a set of closed horizontal unit line segments in the plane that represents G. The location of each bar A in R is uniquely determined by the x and y coordinates of its left endpoint, which we will denote by x(A) and y(A), respectively. Theorem 2 For every k ≥ 0 the graph K3k+3 is a unit bar k-visibility graph, and is the largest complete unit bar k-visibility graph. MORE DIRECTIONS IN VISIBILITY GRAPHS 57 Proof: Let G be a complete graph with n vertices, and let R be a unit bar k-visibility representation of G. We show that n ≤ 3k + 3. Suppose that A and B are the two bars in R that have the smallest and largest y-values, respectively. In other words, y(A) < y(C) < y(B) for all C other than A and B in R. Without loss of generality we may assume that bars in R have distinct x-coordinates. Therefore we may also assume by symmetry that x(A) < x(B). We partition the remaining bars in R into three types. We say a bar C in R is type I if x(C) < x(A), type II if x(A) < x(C) < x(B), and type III if x(B) < x(C), as shown in Figure 1.