convex grid drawings of plane graphs with rectangular contours akira kamada (tohoku university)...
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Convex Grid Drawings of Plane Graphswith Rectangular Contours
Akira Kamada (Tohoku University)Kazuyuki Miura (Fukushima University)Takao Nishizeki (Tohoku University)
no edge-intersection
Drawings
1 23
84 5
6
7910 11
12
12
3
8
4
5
6
79
10
11
12
12
3
8
4
5
6
7 9
10
11
12
13
84 5
6
7 9
1011
12
2
plane graph
convex grid drawing
vertex
edge
Convex grid drawing
1 23
84 5
6
7910 11
12
12
3
8
4
5
6
7 9
10
11
12
12
3
8
4
5
6
79
10
11
12
vertex : grid point
edge : straight line segment without edge-intersection
face : convex polygon
plane graph
vertex
edge
outer face
Known results (1)plane graph
2-connected
internally 3-connected
convex grid drawing
known results (1)
plane graph
2-connected
Known results (1)
u
separation pair
inner vertex
cannot be simultaneouslydrawn as convex polygons
outer vertex
v
plane graph
2-connected
outer vertex
Known results (1)
cannot be simultaneouslydrawn as convex polygons
no outer vertexu
v
u
inner vertex
v
Known results (1)plane graph
2-connected
internally 3-connected
v
u
v
for any separation pair { u , v } of G , 1) both u and v are outer vertices, and 2) each component of G – { u , v } contains an outer vertex
u
v
inner vertex
no outer vertex
u
3-connected
u
Known results (1)plane graph
2-connected
internally 3-connected
convex grid drawing
known results (1)
internally 3-connected
leaves 4
leaves 3
Known results (2)
convex grid drawing
decomposition tree
known results (2)
n n size
our result ?
leaves 4 : open problem
polynomial size
Algorithminner convex grid drawing
internally 3-connected graphs
leaves 2
U10
U9
U8
U7
U6
U5U4 U2
U1
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
U3
new “shift method”
Canonical decomposition
U10
U9
U8
U7
U6
U5U4 U2
U1
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
U3
G5
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U6
U5U4
U3
U2
U1
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
G8
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U9
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
Algorithm
U10
U9
U8
U7
U6
U5U4
U3
U2
U1
convex polygon①
slope 0 or 1②
convex apex has upper neighbors③
new “shift method”
concave apex
Algorithm
less than 180
convex apex has upper neighbors③
new “shift method”U10
U9
U8
U7
U6
U5U4
U3
U2
U1
convex apex
G1
Algorithm
convex polygon①
slope 0 or 1②
convex apex has upper neighbors③
new “shift method”
U2
shift shift
G1
Algorithm
U2
convex polygon①
slope 0 or 1②
convex apex has upper neighbors③
new “shift method”
shift shift
G92n1
Algorithm
n1
convex polygon①
slope 0 or 1②
convex apex has upper neighbors③
new “shift method”
2n1
Algorithm
n1
new “shift method”
n12
inner convex grid drawing
internally 3-connected graphs
leaves 2
2n1
Algorithm
n12
2n2n2n2
2
inner convex grid drawings
new “shift method”
internally 3-connected graphs
leaves 2
n1
2n1
Algorithm
n12
inner convex grid drawings
max{ 2n1 , 2n2 }
n22
2nn n1 n2
internally 3-connected graphs
leaves 2
2n2n2
n1
2n
Algorithm
every inner face isa convex polygon
n12
n22
inner convex grid drawings
internally 3-connected graphs
leaves 2
n2
n1
2n
Algorithminner convex grid drawings
n12
n22
| slope | 1
internally 3-connected graphs
leaves 2
n2
n1
2n
Algorithminner convex grid drawings
n12
n22
| slope | 1
internally 3-connected graphs
leaves 2
n2
n1
2n
Algorithmconvex grid drawinginner convex grid drawings
2n
n12
n22
| slope | 1
edge-intersection
non-convex polygon
2n
Algorithmconvex grid drawinginner convex grid drawings
2n
n12
n22
2n1
| slope | 1 1 | slope |
no edge-intersection, and
every face newly createdis a convex polygon
Conclusionsleaves 4 convex grid drawing
2n
n2
linear time
n vertices
degree 4
3-connected graph
ring degree 4
Open probleminternally 3-connected
leaves 3
convex grid drawing
leaves 4
known results (2) our result
2n n2 size
leaves 5 : open problem
G1
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U2
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
G2
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U3
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
G3
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U4
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
G4
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U5
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
G5
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U6
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
G6
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U7
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
G7
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U8
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
G8
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U9
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
G9
Canonical decomposition
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
Gk is induced byU1 U2 ∙∙∙ Uk
U10 = Um
Uktwo or more
one or more neighbors
Gk1
Uk
one or more(each vertex)
Gk1
exactly one(leftmost and rightmost vertices)
Canonical decomposition
U10
U9
U8
U7
U6
U5U4
U3
U2
U1
canonical decomposition
= ( U1 , U2 , ∙∙∙ , Um1 , Um )
2n
2n × n2 sizeconvex grid drawing
2n1
n12
n22
n2
n12 n2
2 2n1
( n1 n2 )2 2n1n2 2n1
n2
n n1 n2
n1 , n2 5