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 2010August 11, 2010

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

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

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

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

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

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

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

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

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

Is Draw r (a y, b y, c y ) = True ? Previous ResultsProblem Formulation CCCG 2010 August 11, 2010 a b e c k l Representative vertex e 133 c b a No line is available to place the representative vertex e Draw e (1, 2, 2) = False k l e a b c 11 Draw e (1, 3, 3) = True

k l e a b c Previous ResultsRecursive Solution CCCG 2010 August 11, 2010 Draw r (a y, b y, c y ) = False, if max{a y, b y, c y } − min{a y, b y, c y } = 0. 12

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

Draw r (a y, b y, c y ) = True, if max{a y, b y, c y } − min{a y, b y, c y } ≥ 1 and r is a dummy vertex. and r is a dummy vertex. Previous ResultsRecursive Solution CCCG 2010 August 11, 2010 c b a Draw r (a y, b y, c y ) = False, if max{a y, b y, c y } − min{a y, b y, c y } = 1 and r is an internal vertex. and r is an internal vertex. a b c Draw r (a y, b y, c y ) = False, if max{a y, b y, c y } − min{a y, b y, c y } = 0. 14

Previous ResultsRecursive Solution CCCG 2010 August 11, 2010 Draw r (a y, b y, c y ) = True, if max{a y, b y, c y } − min{a y, b y, c y } ≥ 1 and r is a dummy vertex. and r is a dummy vertex. Draw r (a y, b y, c y ) = False, if max{a y, b y, c y } − min{a y, b y, c y } = 1 and r is an internal vertex. and r is an internal vertex. Draw r (a y, b y, c y ) = False, if max{a y, b y, c y } − min{a y, b y, c y } = 0. r a b c Draw r (a y, b y, c y ) = v r y {Draw r (a y, b y, r y ) & Draw r (b y, c y, r y ) & Draw r (c y, a y, r y )}, otherwise. O(1) h O(h) 15

Previous ResultsComplexity Analysis CCCG 2010 August 11, 2010 Draw r (a y, b y, c y ) h O(h)... O(h) O(n) ×× × = O(nh 3 ) O(nh 3 ) × O(h) = O(nh 4 ) O(nh 4 ) × O(h min ) = O(nh 5 min ) h min Computation of each entry is obtained in O(h) time. O( nh 4 min ) 16

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

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

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

Future Works CCCG 2010 August 11, 2010 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. 20

CCCG 2010 August 11, 2010