Graphs with large total angular resolution Oswin Aichholzer Matias - PowerPoint PPT Presentation

W I S S E N T E C H N I K L E I D E N S C H A F T Graphs with large total angular resolution Oswin Aichholzer Matias Korman Yoshio Okamoto Irene Parada Daniel Perz Andr´ e van Renssen Birgit Vogtenhuber 18. September 2019 www.tugraz.at

Introduction 2 Definition Definition (Total angular resolution) The total angular resolution of a straight-line drawing is the minimum angle between two intersecting edges of the drawing. The total angular resolution of a graph G , or short TAR( G ) , is the maximum total angular resolution over all straight-line drawings of this graph. 18. September 2019

Introduction 3 Motivation Crossing resolution Angular resolution Total angular resolution 18. September 2019

Introduction 4 Considered questions Can we find an upper bound for the number of edges of graphs G with TAR( G ) > 60 ◦ ? What is the complexity of deciding whether TAR( G ) ≥ 60 ◦ ? 18. September 2019

Introduction 5 Upper bounds for the number of edges Number of edges of drawings with: crossing resolution 90 ◦ : ≤ 4 n − 10 [Didimo, Eades, Liotta, 2011] crossing resolution greater than 60 ◦ : ≤ 6 . 5 n − 10 [Ackermann, Tardos, 2007] total angular resolution greater than 60 ◦ : ≤ 2 n − 6 with some small exceptions [This work] 18. September 2019

Upper bound for the number of edges 6 Planarized drawing Planarized drawing: replace every crossing by a vertex. 18. September 2019

Upper bound for the number of edges 6 Planarized drawing Planarized drawing: replace every crossing by a vertex. 18. September 2019

Upper bound for the number of edges 7 Size of a cell Size of a cell: number of sides in planarized drawing incident to this cell. C 18. September 2019

Upper bound for the number of edges 7 Size of a cell Size of a cell: number of sides in planarized drawing incident to this cell. C Size of cell C : 4 18. September 2019

Upper bound for the number of edges 7 Size of a cell Size of a cell: number of sides in planarized drawing incident to this cell. C Size of cell C : 6 18. September 2019

Upper bound for the number of edges 8 Basic idea Let D be a drawing. If TAR( D ) > 60 ◦ , then D does not contain a triangle and no three edges cross in one point. So every cell has at least size 4. 18. September 2019

Upper bound for the number of edges 8 Basic idea Let D be a drawing. If TAR( D ) > 60 ◦ , then D does not contain a triangle and no three edges cross in one point. So every cell has at least size 4. Lemma Given a connected drawing D with n ≥ 1 vertices and m edges. The unbounded cell of D has size k and TAR( D ) > 60 ◦ . Then m ≤ 2 n − 2 − ⌈ k / 2 ⌉ . 18. September 2019

Upper bound for the number of edges 9 m ≤ 2 n − 4 Lemma Given a drawing D with TAR( D ) > 60 ◦ . If the unbound cell has size at least 4 , then m ≤ 2 n − 4 . The only possible triangle-free drawings with an unbound cell of size at most 2 are: the empty graph a single vertex two vertices joined by an edge. 18. September 2019

Upper bound for the number of edges 10 Idea to continue D’ 18. September 2019

Upper bound for the number of edges 10 Idea to continue m ′ ≤ 2 n ′ − 4 m ′ ≥ m − 8 m ≤ 2 n − 6 n ′ = n − 5 D’ 18. September 2019

Upper bound for the number of edges 11 Exceptions E3 E0 E1 E2 E4 E5 E6 E7 E8 E9 18. September 2019

Upper bound for the number of edges 12 Result Theorem Given a graph G with TAR( G ) > 60 ◦ . Then m ≤ 2 n − 6 or G is in the exceptions. 18. September 2019

Upper bound for the number of edges 13 Tightness Drawing of a graph with TAR( G ) > 60 ◦ and 2 n − 6 edges. 18. September 2019

Upper bound for the number of edges 13 Tightness Drawing of a graph with TAR( G ) > 60 ◦ and 2 n − 6 edges. 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 14 Hardness results Before: It is NP-hard to decide whether a graph G has angular resolution ≥ 90 ◦ . [Forman et al. 1993] 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 14 Hardness results Before: It is NP-hard to decide whether a graph G has total angular resolution ≥ 90 ◦ . [Forman et al. 1993] 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 14 Hardness results Before: It is NP-hard to decide whether a graph G has total angular resolution ≥ 90 ◦ . [Forman et al. 1993] Theorem It is NP -hard to decide whether a graph G has TAR( G ) ≥ 60 ◦ . 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 14 Hardness results Before: It is NP-hard to decide whether a graph G has total angular resolution ≥ 90 ◦ . [Forman et al. 1993] Theorem It is NP -hard to decide whether a graph G has TAR( G ) ≥ 60 ◦ . Proof by reduction from 3SAT. 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 15 Construction ℓ 2 ℓ 1 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 16 Variable gadgets X 1 ℓ 2 ℓ 1 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 16 Variable gadgets X 1 ℓ 2 x 1 , 4 x 1 , 4 x 1 , 3 x 1 , 3 x 1 , 2 x 1 , 2 x 1 , 1 x 1 , 1 ℓ 1 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 16 Variable gadgets X 1 ℓ 2 x 1 , 4 x 1 , 4 x 1 , 3 x 1 , 3 x 1 , 2 x 1 , 2 x 1 , 1 x 1 , 1 ℓ 1 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 17 Clause gadget C 4 X 1 ℓ 2 x 1 , 4 x 1 , 4 C 3 x 1 , 3 x 1 , 3 C 2 x 1 , 2 x 1 , 2 x 1 , 1 x 1 , 1 ℓ 1 C 1 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 17 Clause gadget X 1 ℓ 2 x 1 , 4 x 1 , 4 x 1 , 3 x 1 , 3 x 1 , 2 x 1 , 2 x 1 , 1 x 1 , 1 ℓ 1 C 1 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 18 Connections Connection to: left side of variable gadget right side of variable gadget C j C j X i,j X i,j X i,j X i,j C j X i,j X i,j 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 18 Connections ℓ 2 not possible with ≥ 60 ◦ ℓ 1 C 1 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 18 Connections ℓ 2 not possible with ≥ 60 ◦ ℓ 1 C 1 18. September 2019

NP-hardness for TAR( G ) ≥ 60 ◦ 19 Example ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 4 X 1 ℓ 2 C 3 X 2 C 2 X 3 ℓ 1 C 1 18. September 2019

Open problems 20 Open problems Do almost all graphs k 90 ◦ have at most 2 n − 2 −⌊ k with TAR( G ) > k − 2 2 ⌋ edges? At which angle(s) α does the decision problem, whether TAR( G ) ≥ α , change from NP-hard to polynomially solvable? 18. September 2019