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Convex Grid Drawings of Plane Graphs
with Rectangular Contours Akira Kamada (Tohoku University) Kazuyuki Miura (Fukushima University) Takao Nishizeki (Tohoku University)
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Drawings vertex edge plane graph no edge-intersection
1 2 3 8 4 5 6 7 9 10 11 12 1 2 3 8 4 5 6 7 9 10 11 12 vertex edge plane graph 1 2 3 8 4 5 6 7 9 10 11 12 no edge-intersection 1 3 8 4 5 6 7 9 10 11 12 2 convex grid drawing
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Convex grid drawing vertex edge plane graph vertex ： grid point
1 2 3 8 4 5 6 7 9 10 11 12 1 2 3 8 4 5 6 7 9 10 11 12 vertex edge plane graph 1 2 3 8 4 5 6 7 9 10 11 12 vertex ： grid point edge ： straight line segment without edge-intersection face ： convex polygon outer face
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Known results (1) plane graph 2-connected internally 3-connected
convex grid drawing
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Known results (1) plane graph no cut vertex cut vertex 2-connected
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Known results (1) plane graph 2-connected inner vertex u v
separation pair outer vertex cannot be simultaneously drawn as convex polygons
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Known results (1) plane graph 2-connected inner vertex no outer vertex
cannot be simultaneously drawn as convex polygons
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Known results (1) plane graph 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 2-connected inner vertex internally 3-connected u u u v u v v 3-connected no outer vertex
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Known results (1) plane graph 2-connected internally 3-connected
convex grid drawing
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Decomposition tree
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Decomposition tree separation pair
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Decomposition tree virtual edge
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Decomposition tree
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Decomposition tree repeat this operation
until no more splits are possible
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Decomposition tree no more splits are possible
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Decomposition tree triangle merge triangle no more splits are possible
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Decomposition tree ring (cycle) merge no more splits are possible
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Decomposition tree : leaf
3-connected component decomposition tree [HT73] 3-connected component : leaf
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Known results (2) leaves 3 convex grid drawing triangular contour n
[CK97], [MAN05] triangular contour n n n vertices
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? Known results (2) internally 3-connected decomposition tree
leaves 4 : open problem leaves 4 leaves 3 ? known results (2) our result convex grid drawing polynomial size n n size
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Our result rectangular contour leaves 4 convex grid drawing n2
n vertices 2n
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Our result rectangular contour leaves 4 convex grid drawing
corner vertex
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Our result rectangular contour leaves 4 convex grid drawing
triangular contour corner vertex
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Our result rectangular contour leaves 4 convex grid drawing
corner vertex
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Our result leaves 4 convex grid drawing
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internally 3-connected graphs
Main idea internally 3-connected graphs leaves 2 leaves 4 divide
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internally 3-connected graphs
Main idea internally 3-connected graphs leaves 2 inner convex grid drawings new “shift method”
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Main idea inner convex grid drawings convex grid drawing
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Main idea input graph convex grid drawing n2 n vertices 2n
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internally 3-connected graphs
Algorithm internally 3-connected graphs leaves 2 leaves 4 divide
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internally 3-connected graphs
Algorithm internally 3-connected graphs leaves 2 leaves 4 divide
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internally 3-connected graphs
Algorithm internally 3-connected graphs leaves 2 inner convex grid drawings new “shift method”
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internally 3-connected graphs
Algorithm internally 3-connected graphs leaves 2 inner convex grid drawing U10 U9 U8 U7 U6 U5 U4 U2 U1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) U3 new “shift method”
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Canonical decomposition
U10 U9 U8 U7 U6 U5 U4 U2 U1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) U3
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Canonical decomposition
P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Um U1
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) U6 U5 U4 U3 U2 U1 G5 Gk is induced by U1 U2 ∙∙∙ Uk
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) U9 Gk is induced by U1 U2 ∙∙∙ Uk G8
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ U10 new “shift method” U6 U9 U4 U2 U5 U3 U8 U7 U1
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Algorithm inner face ① convex polygon new “shift method” U10 U6 U9 U4
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Algorithm ② slope 0 or 1 new “shift method” U10 U6 U9 U4 U2 U5 U3
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Algorithm convex apex has upper neighbors ③ convex apex less than 180
concave apex U10 new “shift method” U6 U9 U4 U2 U5 U3 U8 U7 U1
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” U1
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” U2 G1 shift shift
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” U2 G1 shift shift
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G2 U3
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G2 U3
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” U4 G3
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” U4 G3
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G4 U5
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G4 U5
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” U6 G5
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” U6 G5
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G6 U7
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G6 U7
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G7 U8
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G7 U8
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G8 U9
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G8 U9
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Algorithm convex polygon ① slope 0 or 1 ②
convex apex has upper neighbors ③ new “shift method” G9 n1 2n1
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Grid size G4 U5 n12 G3 U4 n1 G2 U3 G1 U2 U1 2n1
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internally 3-connected graphs
Algorithm internally 3-connected graphs leaves 2 inner convex grid drawing new “shift method” n1 n12 2n1
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internally 3-connected graphs
Algorithm internally 3-connected graphs leaves 2 inner convex grid drawings n2 2n2 n22 new “shift method” n1 n12 2n1
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internally 3-connected graphs
Algorithm internally 3-connected graphs leaves 2 inner convex grid drawings n2 2n2 n22 max{ 2n1 , 2n2 } n n1 n2 2n n1 n12 2n1
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internally 3-connected graphs
Algorithm internally 3-connected graphs leaves 2 inner convex grid drawings n2 n22 every inner face is a convex polygon n1 n12 2n
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internally 3-connected graphs
Algorithm internally 3-connected graphs leaves 2 inner convex grid drawings n2 n22 | slope | 1 n1 n12 2n
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internally 3-connected graphs
Algorithm internally 3-connected graphs leaves 2 inner convex grid drawings n2 n22 | slope | 1 n1 n12 2n
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Algorithm inner convex grid drawings convex grid drawing n22
non-convex polygon | slope | 1 edge-intersection n12 2n 2n
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Algorithm inner convex grid drawings convex grid drawing n22 2n1
| slope | 1 n12 2n 2n
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Algorithm inner convex grid drawings convex grid drawing n22 2n1
| slope | 1 1 | slope | n12 2n 2n
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Algorithm inner convex grid drawings convex grid drawing n22 2n1
| slope | 1 1 | slope | no edge-intersection, and every face newly created is a convex polygon n12 2n 2n
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Algorithm inner convex grid drawings convex grid drawing n2 n22 2n1
H = n12 2n1 n22 n2 n12 2n 2n
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Conclusions leaves 4 convex grid drawing n2 linear time n vertices
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Conclusions convex grid drawing 2n n2 linear time n vertices
leaves 4 3-connected graph degree 4 ring degree 4
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Conclusions 4n 2n 2n1 2n1 2n2 n vertices ring degree 4
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Conclusions 4n 2n 2n1 2n1 2n2 n vertices degree 3 degree 3
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Conclusions degree 3 degree 3 4n 2n 2n1 2n1 2n2 n vertices
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Open problem internally 3-connected leaves 5 : open problem
known results (2) our result convex grid drawing 2n n2 size
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Open problem leaves 5 convex grid drawing ? n vertices ?
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Open problem leaves 5 convex grid drawing ? n vertices ?
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Grid size U7 n1 U6 U5 U4 G6 U2 U3 n12 U1 G5 G4 G3 G2 G1 2n1
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Decomposition tree separation pair
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Decomposition tree virtual edge
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Decomposition tree
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Decomposition tree repeat this operation
until no more splits are possible
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Decomposition tree repeat this operation
until no more splits are possible
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Decomposition tree repeat this operation
until no more splits are possible
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Decomposition tree repeat this operation
until no more splits are possible
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Decomposition tree repeat this operation
until no more splits are possible
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Decomposition tree repeat this operation
until no more splits are possible
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Decomposition tree triangle merge triangle no more splits are possible
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Decomposition tree ring(cycle) merge
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Decomposition tree : leaf
3-connected component decomposition tree [HT73] : leaf
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Canonical decomposition
P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Um U1
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) U2 Gk is induced by U1 U2 ∙∙∙ Uk G1
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) G2 U3 Gk is induced by U1 U2 ∙∙∙ Uk
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) U4 G3 Gk is induced by U1 U2 ∙∙∙ Uk
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) G4 U5 Gk is induced by U1 U2 ∙∙∙ Uk
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) U6 G5 Gk is induced by U1 U2 ∙∙∙ Uk
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) G6 Gk is induced by U1 U2 ∙∙∙ Uk U7
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) G7 Gk is induced by U1 U2 ∙∙∙ Uk U8
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) G8 U9 Gk is induced by U1 U2 ∙∙∙ Uk
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Canonical decomposition
Uk two or more one or more neighbors Gk1 canonical decomposition P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) Uk one or more (each vertex) Gk1 exactly one (leftmost and rightmost vertices) U10 = Um G9 Gk is induced by U1 U2 ∙∙∙ Uk
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Canonical decomposition
P = ( U1 , U2 , ∙∙∙ , Um1 , Um ) U10 U9 U8 U7 U6 U5 U4 U3 U2 U1
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2n × n2 size convex grid drawing n22 n12 n22 2n1
( n1 n2 )2 2n1n2 2n1 n2 n n1 n2 n1 , n2 5 n12 2n
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