Bar Visibility Research Articles

Sliding column model for t-unit bar visibility representations of graphs

A bar visibility representation of a graph [Formula: see text] is an assignment of the vertices of [Formula: see text] to distinct horizontal line segments in the plane so that two vertices are adjacent in [Formula: see text] if and only if there is an uninterrupted vertical channel of positive width that joins the bars corresponding to those vertices. A bar visibility representation is called a unit bar visibility representation if every bar has the same length. If each vertex is assigned to [Formula: see text] distinct bars of the same length in a unit bar visibility representation then the representation is called [Formula: see text]-unit bar visibility representation. In this paper, we introduce a “sliding column model” for [Formula: see text]-unit bar visibility representation and show that every graph of maximum degree [Formula: see text] has a [Formula: see text]-unit bar visibility representation for [Formula: see text]. We also show that a planar graph of maximum degree 3 having [Formula: see text] vertices and [Formula: see text] edges has a 2-unit bar visibility representation on [Formula: see text] columns and a 3-connected cubic graph of [Formula: see text] vertices admits a 2-unit bar visibility representation on [Formula: see text] columns.

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 $k$- and rectangle visibility graphs to form rectangle $k$-visibility graphs (R$k$VGs), and further generalize these to higher dimensions. A graph is a $d$-dimensional R$k$VG 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 R$k$VG. 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 $K_m$ is at most $\lceil{m/(2(k+1))}\rceil,$ of complete $r$-partite graphs is at most $r+1,$ and of the $m^{\rm th}$ hypercube graph $Q_m$ is at most $\lceil{2m/3}\rceil$ in general, and at most $\lfloor{\sqrt{m}\,}\rceil$ for $k=0,~ m \ne 2.$

On the Bar Visibility Number of Complete Bipartite Graphs

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 6 May 2019Accepted: 07 June 2021Published online: 28 September 2021Keywordsbar visibility number, bar visibility graph, planar graph, thickness, complete bipartite graphAMS Subject Headings05C62, 05C10Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

Upper bounds for bar visibility of subgraphs and [formula omitted]-vertex graphs

A t-bar visibility representation of a graph G assigns each vertex up to t horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive width. The least t such that G has a t-bar visibility representation is the bar visibility number of G, denoted by b(G). We show that if H is a spanning subgraph of G, then b(H)≤b(G)+1. It follows that b(G)≤⌈n∕6⌉+1 when G is an n-vertex graph. This improves the upper bound obtained by Chang et al. (2004).

S-visibility problem in VLSI chip design

Abstract: In this paper, a new version of very large scale integration (VLSI) layouts compaction problem is considered. Bar visibility graph (BVG) is a simple geometric model for VLSI chip design and layout problems. In all previous works, vertical bars or other chip components in the plane model gates, as well as edges, are modeled by horizontal visibilities between bars. In this study, for a given set of vertical bars, the edges can be modeled with orthogonal paths known as staircases. Therefore, we consider a new version of bar visibility graphs (BsVG). We then present an algorithm to solve the s-visibility problem of vertical segments, which can be implemented on a VLSI chip. Our algorithm determines all the pairs of segments that are s-visible from each other.

The Partial Visibility Representation Extension Problem

For a graph G, a function psi is called a bar visibility representation of G when for each vertex v in V(G), psi (v) is a horizontal line segment (bar) and uv in E(G) if and only if there is an unobstructed, vertical, varepsilon -wide line of sight between psi (u) and psi (v). Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph G, a bar visibility representation of G, additionally, puts the bar psi (u) strictly below the bar psi (v) for each directed edge (u, v) of G. We study a generalization of the recognition problem where a function psi ' defined on a subset V' of V(G) is given and the question is whether there is a bar visibility representation psi of G with psi (v) = psi '(v) for every v in V'. We show that for undirected graphs this problem, and other closely related problems, is mathsf {NP}-complete, but for certain cases involving directed graphs it is solvable in polynomial time.

S-visibility problem in VLSI chip design

In this paper, a new version of very large scale integration (VLSI) layouts compaction problem is considered. Bar visibility graph (BVG) is a simple geometric model for VLSI chip design and layout problems. In all previous works, vertical bars or other chip components in the plane model gates, as well as edges, are modeled by horizontal visibilities between bars. In this study, for a given set of vertical bars, the edges can be modeled with orthogonal paths known as staircases. Therefore, we consider a new version of bar visibility graphs (BsVG). We then present an algorithm to solve the s-visibility problem of vertical segments, which can be implemented on a VLSI chip. Our algorithm determines all the pairs of segments that are s-visible from each other.

Bar Visibility Numbers for Hypercubes and Outerplanar Digraphs

A t-bar visibility representation of a graph G assigns each vertex a union of at most t horizontal segments ("bars") in the plane so that vertices u and v are adjacent if and only if some bar assigned to u and some bar assigned to v are joined by an unobstructed vertical line of sight having positive width. For an oriented graph G, having an edge uv requires visibility from a bar for u upward to a bar for v. The visibility number of a graph or digraph G, written b(G), is the least t such that G has a t-bar visibility representation. For the n-dimensional hypercube $$Q_n$$Qn, Euler's Formula yields $$b(Q_n)\ge \lceil \frac{n+1}{4}\rceil $$b(Qn)??n+14?. To prove equality, we decompose $$Q_{4k-1}$$Q4k-1 explicitly into k spanning subgraphs whose components have the form [InlineEquation not available: see fulltext.]. When G is an outerplanar oriented graph, always $$b(G)\le 2$$b(G)≤2; we give a forbidden substructure characterization for those with $$b(G)=1$$b(G)=1.

Simultaneous visibility representations of plane st-graphs using L-shapes

Let 〈Gr,Gb〉 be a pair of plane st-graphs with the same vertex set V. A simultaneous visibility representation with L-shapes of 〈Gr,Gb〉 is a pair of bar visibility representations 〈Γr,Γb〉 such that, for every vertex v∈V, Γr(v) and Γb(v) are a horizontal and a vertical segment, respectively, which share an end-point. In other words, every vertex is drawn as an L-shape, every edge of Gr is a vertical visibility segment, and every edge of Gb is a horizontal visibility segment. Also, no two L-shapes intersect each other. An L-shape has four possible rotations, and we assume that each vertex is given a rotation for its L-shape as part of the input. Our main results are: (i) a characterization of those pairs of plane st-graphs admitting such a representation, (ii) a quadratic time algorithm to recognize them, and (iii) a linear time drawing algorithm if the test is positive. As an application, starting from a simultaneous visibility representation with L-shapes, we show how to compute a simultaneous embedding of the two graphs with at most two bends per edge and right-angle crossings.

The Unit Bar Visibility Number of a Graph

A $t$-unit-bar representation of a graph $G$ is an assignment of sets of at most $t$ horizontal unit-length segments in the plane to the vertices of $G$ so that (1) all of the segments are pairwise nonintersecting, and (2) two vertices $x$ and $y$ are adjacent if and only if there is a vertical channel of positive width connecting a segment assigned to $x$ and a segment assigned to $y$ that intersects no other segment. The unit bar visibility number of a graph $G$, denoted $ub(G)$, is the minimum $t$ such that $G$ has a $t$-unit-bar visibility representation. Our results include a linear time algorithm that determines $ub(T)$ when $T$ is a tree, bounds on $ub(K_{m,n})$ that determine $ub(K_{m,n})$ asymptotically when $n$ and $m$ are asymptotically equal, and bounds on $ub(K_n)$ that determine $ub(K_n)$ exactly when $n\equiv 1,2\pmod 6$

Linear bounds on matrix extremal functions using visibility hypergraphs

The 0–1 matrix A contains a 0–1 matrix M if some submatrix of A can be transformed into M by changing some ones to zeros. If A does not contain M, then A avoids M. Let ex(n,M) be the maximum number of ones in an n×n 0–1 matrix that avoids M, and let exk(m,M) be the maximum number of columns in a 0–1 matrix with m rows that avoids M and has at least k ones in every column. A method for bounding ex(n,M) by using bounds on the maximum number of edges in bar visibility graphs was introduced in Fulek (2009). By using a similar method with bar visibility hypergraphs, we obtain linear bounds on the extremal functions of other forbidden 0–1 matrices.

Bar k-Visibility Graphs

Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a one-to-one correspondence between vertices of G and bars in S, 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. If bars are allowed to see through each other, the graphs representable in this way are precisely the interval graphs. We consider representations in which bars are allowed to see through at most k other bars. Since all bar visibility graphs are planar, we seek measurements of closeness to planarity for bar k-visibility graphs. We obtain an upper bound on the number of edges in a bar k-visibility graph. As a consequence, we obtain an upper bound of 12 on the chromatic number of bar 1-visibility graphs, and a tight upper bound of 8 on the size of the largest complete bar 1-visibility graph. We also consider the thickness of bar k-visibility graphs, obtaining an upper bound of 4 when k = 1, and a bound that is quadratic in k for k > 1.

Further Results on Bar k-Visibility Graphs

A bar visibility representation of a graph G is a collection of horizontal bars in the plane corresponding to the vertices of G such that two vertices are adjacent if and only if the corresponding bars can be joined by an unobstructed vertical line segment. In a bar k-visibility graph, two vertices are adjacent if and only if the corresponding bars can be joined by a vertical line segment that intersects at most k other bars. Bar k-visibility graphs were introduced by Dean et al. [J. Graph Algorithms Appl., 11 (2007), pp. 45–59]. In this paper, we present sharp upper bounds on the maximum number of edges in a bar k-visibility graph on n vertices and the largest order of a complete bar k-visibility graph. We also discuss regular bar k-visibility graphs and forbidden induced subgraphs of bar k-visibility graphs.

Characterization of $[1,k]$-Bar Visibility Trees

A unit bar-visibility graph is a graph whose vertices can be represented in the plane by disjoint horizontal unit-length bars such that two vertices are adjacent if and only if there is a unobstructed, non-degenerate, vertical band of visibility between the corresponding bars. We generalize unit bar-visibility graphs to $[1,k]$-bar-visibility graphs by allowing the lengths of the bars to be between $1/k$ and $1$. We completely characterize these graphs for trees. We establish an algorithm with complexity $O(kn)$ to determine whether a tree with $n$ vertices has a $[1,k]$-bar-visibility representation. In the course of developing the algorithm, we study a special case of the knapsack problem: Partitioning a set of positive integers into two sets with sums as equal as possible. We give a necessary and sufficient condition for the existence of such a partition.

The Bar Visibility Number of a Graph

The bar visibility number of a graph G, denoted b(G), is the minimum t such that G can be represented by assigning each vertex x the set Sx of points in at most t horizontal segments in the plane so that uv $\in$ E(G) if and only if some point of Su sees some point of Sv via a vertical segment of positive width unobstructed by assigned points. Among our results are the following: (1) Every planar graph has bar visibility number at most 2, which is sharp. (2) $r\le b(K_{m,n})\le r+1$, where $r=\big\lceil\frac{mn+4}{2m+2n}\big\rceil$. (3) $b(K_n)=\lceil{n/6}\rceil$. (4) If G has n vertices, then $b(G)\le \lceil{n/6}\rceil+2$.

Dissociation of orientation discrimination from form detection for motion-defined bars and luminance-defined bars: Effects of dot lifetime and presentation duration

A stationary bar-shaped area that was perfectly camouflaged within a dot pattern was rendered visible by moving the dots inside and outside the bar at equal and opposite speeds. Orientation discrimination for this motion-defined (MD) bar was compared with orientation discrimination for a luminance-defined (LD) bar created by switching off all dots outside the bar. The best values of orientation discrimination threshold were similar for MD and LD bars at long presentation durations and long dot lifetimes. But, as presentation duration or dot lifetime was reduced below 1.0 sec, orientation discrimination threshold for MD bars increased at an accelerating rate, while discrimination for LD bars was comparatively unaffected. However, these effects of presentation duration and dot lifetime were largely due to changes in bar visibility. When bar visibility was normalised relative to the relevant bar detection threshold, the effect of presentation duration upon orientation discrimination was abolished for both MD and LD bars, and the effect of dot lifetime was abolished or even reversed. These observations dissociate detection and discrimination for MD and LD form. We suggest that orientation discrimination for MD and LD bars is determined by opponent-orientation mechanisms whose performance is not directly affected by presentation duration, nor degraded by reducing dot lifetime.

Stabilized target visibility as a function of contrast and flicker frequency

Flicker frequency is found to be the main determiner of stabilized target visibility. A luminous bar of 4′ was superimposed on an evenly illuminated field of 1° circular field. The whole target was stabilized. The bar luminance was varied sinusoidally or in a squarewave fashion around a constant mean level at frequencies ranging from 0.4 to 40 Hz. The contrast ratio of the bar to its background varied between 0.06 (threshold) and 10. The bar stayed visible for the longest duration when the bar flicker was 0.4 Hz. Visibility of the bar which was expressed as the duration before its disappearance, declined as a function of modulation frequency until lowest visibility was obtained at 15 Hz. An increase of frequency beyond this point led to no change in bar visibility. Between 0.4 and 15 Hz, contrast of the target played an indeterminant role, improving visibility for one subject, and having no effect on visibility for the other subjects. For flicker rates above 15 Hz, target contrast was the main determinant of visibility. In relative terms, a decrease in flicker frequency leads to longer visibility than an increase in contrast. Square-wave flicker has the same effect as sine-wave flicker.