MinimumArea Drawings of Plane 3 Trees Debajyoti Mondal

Minimum-Area Drawings of Plane 3 -Trees Debajyoti Mondal, Rahnuma Islam Nishat, Md. Saidur Rahman and Md. Jawaherul Alam Graph Drawing and Information Visualization Laboratory Department of Computer Science and Engineering Bangladesh University of Engineering and Technology (BUET) Dhaka – 1000, Bangladesh CCCG 2010 August 11, 2010

Minimum-Area Drawings A Straight-Line Drawing of G A Plane Graph G H=5 H=4 CCCG 2010 W=8 W=6 A Straight-Line Grid-Drawing of G on 8× 5 grid A Straight-Line Grid-Drawing of G on 6× 4 grid 2 August 11, 2010

Minimum-Area Drawings H=5 H=6 A Straight-Line W=7 Drawing of G A Plane Graph G W=6 H=5 H=4 H=5 CCCG 2010 W=8 W=6 A Straight-Line Grid-Drawing of G W=8 on 8× 5 grid A Straight-Line A Minimum-Area Grid-Drawing of G on 6× 4 grid 3 August 11, 2010

Previous Results de Fraysseix et al. [1990] Straight- line grid-drawing of plane graphs with n vertices (2 n− 4)×(n− 2) Schnyder [1990] Straight- line grid-drawing of plane graphs with n vertices (n− 2)×(n− 2) Brandenburg [2004] Straight- line grid-drawing of plane graphs with n vertices (4 n/3) × (2 n/3) Krug and Wagner [2008] Whether a planar graph has a drawing on a given area NP-Complete This Presentation Whether a ‘plane 3 -tree’ has a drawing on a given area P CCCG 2010 4 August 11, 2010

Previous Our Results de Fraysseix et al. [1990] Straight- line grid-drawing of plane graphs with n vertices (2 n− 4)×(n− 2) Schnyder [1990] Straight- line grid-drawing of plane graphs with n vertices (n− 2)×(n− 2) We obtain minimum-area drawings for Brandenburg Straight- line grid-drawing (4 n/3) × (2 n/3) [2004] plane 3 -trees of plane graphs with n vertices in polynomial time Krug and Wagner [2008] Whether a plane graph has a drawing on a given area NP-Complete This Presentation Whether a ‘plane 3 -tree’ has a drawing on a given area P CCCG 2010 5 August 11, 2010

Previous Plane 3 -tree Results b b j k e e d l a o i n l h f g d o a m i f g n h m c c A plane 3 -tree G CCCG 2010 6 August 11, 2010

Properties Previous of Plane Results 3 -trees b The representative vertex of G A plane 3 -tree b j A plane 3 -tree k j c k e e d l a o i n l h f g d o a m i f g n h m c c A plane 3 -tree G A plane 3 -tree The representative vertex of G is the vertex which is neighbor of all the three outer vertices of G. CCCG 2010 7 August 11, 2010

Our Idea. Previous : Dynamic Results Programming b j c k e d l a o i f g n h m c A plane 3 -tree G CCCG 2010 8 August 11, 2010

Let’s Try Previous a Simpler Results Problem c b c a a b a c c b CCCG 2010 9 August 11, 2010

Let’s Try Previous a Simpler Results Problem c c e b a k No line is available to place the representative vertex e e l c k b e l b a a c k e No line is available to place the vertex l Let’s check whether this small plane 3 -tree admits c with this placement of a, bband c or not a drawing a k e l a CCCG 2010 10 August 11, 2010

Problem Previous Formulation Results c 3 2 1 k a e l b Is c k e l a Drawr(ay, by, cy) = True 1 Representative vertex e c b 3 3 ? Drawe(1, 3, 3) = True b Drawe(1, 2, 2) = False a No line is available to place the representative vertex e CCCG 2010 11 August 11, 2010

Recursive Previous Results Solution c k a e l b Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 0. CCCG 2010 12 August 11, 2010

Recursive Previous Results Solution c c k a e l b a b No line is available to place the representative vertex Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 0. Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 1 and r is an internal vertex. CCCG 2010 13 August 11, 2010

Recursive Previous Results Solution c c b a Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 0. Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 1 and r is an internal vertex. Drawr(ay, by, cy) = True , if max{ay, by, cy} − min{ay, by, cy} ≥ 1 and r is a dummy vertex. CCCG 2010 14 August 11, 2010

Recursive Previous Results Solution c Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{O(1) ay, by, cy} = 0. r b a Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 1 O(1) and r is an internal vertex. Drawr(ay, by, cy) = True , if max{ay, by, cy} − min{O(1) ay, by, cy} ≥ 1 and r is a dummy vertex. Drawr(ay, by, cy) = v ry {Drawr(ay, by, ry) & Drawr(by, cy, ry) & Drawr(cy, ay, ry)}, otherwise. h CCCG 2010 15 O(h) August 11, 2010

Complexity Previous Results Analysis . . . h hmin O(nh 4 min) Drawr(ay, by, cy) O(n) × O(h) = O(nh 3) Computation of each entry is obtained in O(h) time. O(nh 3) × O(h) = O(nh 4) × O(hmin) = O(nh 5 min) CCCG 2010 16 August 11, 2010

A minimumarea grid drawing of G A plane 3 -tree G . . Minimum-Area Grid Drawings of Plane 3 -Trees Patch the drawings of the subproblems to obtain the final drawing. . . . false CCCG 2010 17 August 11, 2010

Lower Bound on Area (2 n/3 -1) Nested triangles graph There exist plane graphs with n vertices that takes � 2(n-1)/3�×� 2(n-1)/3� area in any straight-line grid drawing. Frati et al. [2008]: There exist plane graphs with n vertices, n is a multiple of three, that takes (2 n/3 -1) ×(2 n/3) area in any straight-line grid drawing. CCCG 2010 18 August 11, 2010

Lower Bound on Area: � 2(n-1)/3�×� 2(n-1)/3� � 2 n/3 -1�× 2�n/3� When is a multiple Input planen 3 -trees. of three, this bound is the same as the one by Frati et al. Minimum-area grid drawings. We observe that there exist plane 3 -trees with n ≥ 6 vertices that takes � 2 n/3 -1�× 2�n/3� area in any straight-line grid drawing. CCCG 2010 19 August 11, 2010

Future Works Devising a simpler algorithm to obtain minimum area drawings of plane 3 -trees. Determining the minimum area drawings for the other plane graphs with bounded treewidth. Determining the area lower bound of straight-line grid drawings of planar 3 -trees when the outer face is not fixed. CCCG 2010 20 August 11, 2010

CCCG 2010 August 11, 2010