Bar-Visibility
Graphs,
Generalized
an d
Specialized
Alice Dean
Math. & Comp. Sci. Dept.
Skidmore College
SMC/UVM
adean@skidmore.edu
www.skidmore.edu/~adean
Combinatorics
Seminar,
October 29,
2003
Talk Outline
1. Introduction: Definitions, motivation,
history
2. BVGs: Characterization using stnumbering
3. Generalization: BVGs on surfaces
4. Specialization: Unit BVGs
5. Other variations and open problems
Warning
There is a definite prejudice in topic
selection toward those I’ve been
personally involved in!
Page 2 of 31
Introduction:
Visibility Drawings of Graphs

Vertices are represented on a
geometric surface by objects such as
bars, arcs, rectangles, boxes

Edges are represented by
visibilities in specified directions
(horizontal, vertical, radial)
Page 3 of 31
Motivations and Applications
 Modeling tool for digital circuit design
 Edges are implied, rather than
drawn, simplifying complex displays
of complex networks
 Bars or boxes can have data labels
written in them
 Some visibility models can display
nonplanar graphs in the plane
Page 4 of 31
Early Work on Visibility Graphs
1976, Garey, Johnson, and So:
 Use visibility graph models to design
a method of testing printed circuits
Reduces potential of O(n2) tests to
maximum of 12.
1985, Schlag, Luccio, Maestrini, Lee,
Wong:
 Study bar-visibility graphs as models
of stick diagrams used in VLSI
design
1985-86, Wismath, and
independently, Tamassia and Tollis,
also Rosenstiehl and Tarjan:
 Give simple, linear-time
characterization bar-visibility graphs
in the plane
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BVGs in the Plane
Vertex = horizontal bar in R2
Edge IFF unobstructed vertical
visibility band (width > 0)
 Vertical lines (bands of width 0) do
not produce edges
 Bars may be either open or closed
segments
 Easy to see that all BVGs are planar
Page 6 of 31
Characterizing BVGs in the Plane
Theorem [TT, W, RT]. A graph G is a
BVG IFF it has a planar embedding
with all cutpoints on a common face.
 Necessity proof not hard: Every
BVG induces a planar graph with all
cutpoints on outer face.
 Sufficiency proof: Not difficult to
paste layouts of 2-connected blocks
together.
 2-connected sufficiency proof:
Requires idea of st-numbering
Definition: If |V(G)| = n and s and t are
distinct vertices, an st-numbering of G
numbers the vertices 1, …, n so that
1. #(s) = 1 and #(t) = n
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2. Every other vertex has a lowernumbered neighbor and highernumbered neighbor.
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Constructing an st-numbering
Facts:
 If G is 2-connected and s-t is an edge,
then G has an st-numbering [LEC]
 In this case there is an O(n+m)
algorithm to construct numbering [ET]
Heuristic for plane 2-connected
graphs:
 Can always do it provided s, t lie on
common face
t
 Embed G so that s,
t are on outer face,
all faces are
convex (always
possible if G is 2connected), and all
s
vertices at unique
heights with s at bottom, t at top
 Number vertices from bottom to top
Page 9 of 31
2-Connected Proof, Stages 1&2:
st-number G and G*
1. Number vertices;
direct edges:
a) Choose any s and t
on outer face;
b) Give G an stnumbering
c) Direct edges upward,
and add two infinite rays,
one into s, one out of t.
2. Direct dual
edges; number
faces
a) G*-edge
crosses G-edge in
right turn
t 7
6
5
4
3
2
s 1
5
2
4
1
3
Page 10 of 31
6
b) Directed edges induce stnumbering of G*
Page 11 of 31
2-Connected Proof, Stage 3:
Construct BVG Layout
t 7
6
3
5
2
5
4
1
4
6
3
2
s 1
Vertex numbers
correspond to rows
 Bar for vertex k is in
row k
Face numbers
correspond to
columns
 Bar for vertex k
spans columns of
faces that it borders
1
2
3
4
5
6
7
6
5
4
3
2
1
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Generalization: BVGs on Surfaces
 2D surfaces have natural notions of
parallel bars and orthogonal
visibility bands
 Cylinder (86,89,91)
 Sphere (1989)
 Torus (1998)
 Möbius Band (2001)
 Projective Plane (2001)
“Flat” Cylinder, Torus, Möbius Band:
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BVGs for Planar Graphs
on the Cylinder
 Bars parallel to
cylinder axis
 Applications in
VLSI: Linear systolic arrays, bit-slice
architectures
 2-connected case: Very similar to
plane, except there are more “split”
faces, besides outer face
 1-connected case: Hiding blocks is
more difficult now.
Theorem [TT]. A planar graph G is a
cylindrical BVG IFF it has an embedding
whose block-cutface tree is a path IFF
G’s block-cutpoint tree is a caterpillar.
 Planar BVG: Block-cutface tree is a
star
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Other “Planar” Surfaces
 SPHERE: Bars are latitudinal
segments; visibility longitudinal
 Equivalent to BVGs on cylinders with
bars perpendicular to axis
 Closed: Bars can be full circles, v.
Open
Theorem [TT]. G is an open BVG on
the sphere IFF G has a plane
embedding with all cutpoints on
boundary of at most two faces IFF G
has an embedding whose block cutface
tree is a double star.
[TT
Closed Spherical BVGs
]
Open Spherical BVGs
Planar BVGs
Cylindrical BVGs
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Non-planar BVGs on Surfaces:
The Möbius Band
 Characterized by D using generalized
techniques of TT
 Each face
corresponds to
two columns in
layout
 Blocks of “base
graph” laid out in
breadth-first
order
Theorem. A 2-cell embedding of G on
the Möbius band is a BVG (with bars
parallel to axis) IFF (1) the blockcutpoint tree of G is a caterpillar; (2) at
most one block is non-planar; and (3)
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the non-planar block is the “head” of the
caterpillar.
Page 17 of 31
BVGs on Projective Plane
 Bars are
concentric
arcs; visibility
is radial
 As with
sphere, can have open or closed BVG
 Characterized by Hutchinson
Theorem [D]. G is a BVG on the
projective plane IFF G is a BVG on
Möbius band with bars orthogonal to the
axis.
Theorem [Hutchinson]. G is an open
BVG on the projective plane IFF either
(1) G has plane embedding with all but
at most one cutpoint on a common face,
or (2) G has projective plane embedding
with all cutpoints on a common face.
y1
y
a
b
z
b
a
z1
z
y y1
z
z1
b
a
x
x1
x
x1
Page 18 of 31
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
Specialization: Unit BVGs
 My current main topic of study
 MOTIVATION: Circuits in which
components have a single, fixed
size.
Definition: A unit bar-visibility graph
(UBVG) is a BVG in which all bars
have length 1.
Theorem. Kn is a UBVG IFF n  3. Kp,q
is a UBVG IFF p = 1 and q  3, or p =
q = 2.
 Span(U) = # distinct left xcoordinates
E. g., Span = 3:
 Span = 1 IFF G is a
path
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 Span = 2  G is outerplanar
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Cube, Circular Ladders, and Trees
Theorem. The graph K2  Cn is a UBVG
IFF n = 3, or n  6 and even
 Q3 = K2  C4 is not a UBVG
 UBVG is not a hereditary property.
Theorem A tree is a UBVG IFF it has
maximum degree 3 and is a subdivision
of a caterpillar.
 Sufficiency was noted by Hutchinson
and she conjectured necessity
 Necessity requires showing that (a) T
is a UBVG if a subdivision of T is a
UBVG; (b) a certain forbidden
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subgraph for caterpillars is not a
UBVG
Page 22 of 31
UBVG Results: Outerplanar
Graphs
 For a plane graph G, G*I denotes the
dual graph with the outer face deleted.
Theorem. If G is an outerplanar UBVG,
then G*I has max degree 6, and if G is
triangle-free, then G*I has max degree
4.
Theorem. If G is an outerplanar,
triangle-free UBVG, then G*I is a
subdivision of a caterpillar with max
degree 4.
3
1 2 3 4 5 1 2 3 4 5
4
1
2
2
3
5
6
7
4
1
1
2
3
4
5
6
7
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Theorem. If G is a simple, 2-connected,
outerplanar (embedded) graph such that
G*I is a path, then G is a UBVG.
More on structure of the Finite
Dual
If all faces or no faces of G*I are
triangles, structure of G*I is actually
restricted beyond being a caterpillar of
maximum degree 4:
 The neighbors of each face can be
categorized as ‘up,’, ‘down,’ ‘left,’ and
‘right,’
 Each face has at most one neighbor
of each type.
 If G*I is drawn on the integer grid, then
two nodes that lie on same vertical grid
line must lie on a path of faces:
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Possible structure for finite dual
NOT possible structure for finite
dual
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Current Work: Triangulated Polygons
Joint work with E. Gethner (CU Denver)
and J. P. Hutchinson (Macalester C.)
 Big goal: Characterize all subdivided
polygons that are UBVGs
 Smaller goal: Characterize all triangulated
polygons = 2-connected, outerplanar,
near-triangulations
Theorem. If a triangulated polygon G is a
UBVG then its finite dual G*I has maximum
degree 3 and is a subdivision of a
caterpillar.
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Necessity: Forbidden Configurations
Every TPUBVG can be represented
combinatorially in terms of the ‘spine
sequence’ of its finite dual:
Theorem. If G is a TPUBVG, all
subsequences below (and the
corresponding forms with the roles of A and
B interchanged) are ‘essentially’ forbidden
in its spine sequence:
 A A(XB A)iA, i  0, where each XB = B or
NB
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There are several other forbidden
subsequences, but we believe we are very
close to a complete list.
Page 28 of 31
Proof: AAA is forbidden
u2
U2
a2
u1
U1
A1
a3
u3
U3
A2
A3
a1
a4
v
 Each triangle Ui is an ‘up-neighbor’
of Ai, meaning its bars can’t
protrude to left or right of Ai;
 All bars b(ai) are neighbors of bar
b(v), so must lie in corridor of width
< 3;
 The bars b(u1), b(u2), b(u3) need
width 3 or more;
 Contradiction.
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Sufficiency: Layout Algorithm
Step 1: Scan spine sequence for
forbidden subsequences.
Step 2: If OK, lay out UBVG by picturing
graph as lying on triangulated plane, with
some ‘nudges’ (fattened triangles) and
‘wing-splitters’ (subdivided triangles) as
needed
We are hopeful that we will soon:
 Complete the characterization of
triangulated polygons
 Extend the results to quadrangulated
polygons without too much more difficulty
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 Extend results to all subdivided polygons?
To all outerplanar graphs?
Page 31 of 31