On the maximum number of crossings in star-simple drawings of K n - PowerPoint PPT Presentation

On the maximum number of crossings in star-simple drawings of K n with no empty lens Stefan Felsner, Michael Hoffmann, Kristin Knorr , Irene Parada EuroCG2020 18.03.2020, EuroCG 2020

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges Not adjacent edges S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges no crossings no crossings Not adjacent edges S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges no crossings no crossings Not adjacent edges S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges no crossings no crossings Not adjacent edges S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges no crossings no crossings Not adjacent edges S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges no crossings no crossings Not adjacent edges at most 1 S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges no crossings no crossings Not adjacent edges at most 1 S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges no crossings no crossings Not adjacent edges at most 1 S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges no crossings no crossings Not adjacent edges at most 1 S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges no crossings no crossings Not adjacent edges at most 1 S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Simple Drawings Star-simple Drawings Adjacent edges no crossings no crossings Not adjacent edges at most 1 arbitrary many S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 2

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Definition (Maximum Crossing Number) max - cr ( G ) = max (# Crossings in D ( G )) D ( G ) ∈ Simple Drawings † Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol. 8590. 1964. S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 3

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Definition (Maximum Crossing Number) max - cr ( G ) = max (# Crossings in D ( G )) D ( G ) ∈ Simple Drawings Simple Drawings Star-simple Drawings max - cr ∗ ( K n ) max - cr ( K n ) Lower Bound Upper Bound † Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol. 8590. 1964. S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 3

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Definition (Maximum Crossing Number) max - cr ( G ) = max (# Crossings in D ( G )) D ( G ) ∈ Simple Drawings Simple Drawings Star-simple Drawings max - cr ∗ ( K n ) max - cr ( K n ) � n � † Lower Bound 4 � n � † Upper Bound 4 † Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol. 8590. 1964. S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 3

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Simple Drawings vs. Star-simple Drawings Definition (Maximum Crossing Number) max - cr ( G ) = max (# Crossings in D ( G )) D ( G ) ∈ Simple Drawings Simple Drawings Star-simple Drawings max - cr ∗ ( K n ) max - cr ( K n ) � n � † Lower Bound unbounded? 4 � n � † Upper Bound unbounded? 4 † Ringel, Gerhard. ”Extremal problems in the theory of graphs.” Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Vol. 8590. 1964. S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 3

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4

Introduction Structure of Star-simple Drawings Lower and Upper Bounds Summary Restrictions empty lens no empty lens twisting spiraling S. Felsner, M. Hoffmann, K. Knorr , I. Parada Maximum crossing number in star-simple drawings of K n EuroCG2020 4