Boris Klemz, Martin N ollenburg, Roman Prutkin 1) Freie Universit - PowerPoint PPT Presentation

1 2 3 Boris Klemz, Martin N¨ ollenburg, Roman Prutkin 1) Freie Universit¨ at Berlin 2) Vienna University of Technology 3) Karlsruhe Institute of Technology GD2015, Los Angeles Recognizing Weighted Disk Contact Graphs Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Disk Contact Representations A disk contact representation (DCR) of a graph G = ( V, E ) consists of a set D of interior-disjoint disks and a bijection D : V → D such that disks D ( u ) , D ( v ) touch ⇔ ( u, v ) ∈ E . A disk contact graph (DCG) is a graph that can be represented with a DCR. disk contact graph G disk contact representation of G Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Application Areas visualizing statistical data population growth rate, population size, spread of diseases, etc. [Inoue, 2011] DCGs related to Disk Intersection Graphs modeling physical problems e.g., broadcast range of transmitter/ receiver stations covering problems e.g., facility location Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Problem Variations, Results, Related Work All planar graphs are Disk Contact Graphs [Koebe, 1936] We want to have control over disk sizes and the embedding. Weighted DCG Recognition Problem: Does 1 2 3 vertex-weighted graph G have a DCR in which the disks’ radii correspond to their weights ? Embedded DCG Recognition Problem: Does b a c b plane graph G have a DCR with an according c a v rotation system ? v Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Problem Variations, Results, Related Work All planar graphs are Disk Contact Graphs [Koebe, 1936] We want to have control over disk sizes and the embedding. Weighted DCG Recognition Problem: Does 1 2 3 vertex-weighted graph G have a DCR in which the disks’ radii correspond to their weights ? Embedded DCG Recognition Problem: Does b a c b plane graph G have a DCR with an according c a v rotation system ? v star caterpillar tree outerplanar planar Unit-Weighted Breu, Kirkpatrick 98 Unit-Weighted, Embedded Weighted Breu, Kirkpatrick 98 Weighted, Embedded Our results: NP-hardness statements and construction algorithms. Related work: NP-hardness statements and construction algorithms. Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Problem Variations, Results, Related Work All planar graphs are Disk Contact Graphs [Koebe, 1936] We want to have control over disk sizes and the embedding. Weighted DCG Recognition Problem: Does 1 2 3 vertex-weighted graph G have a DCR in which the disks’ radii correspond to their weights ? Embedded DCG Recognition Problem: Does b a c b plane graph G have a DCR with an according c a v rotation system ? v star caterpillar tree outerplanar planar 1. Unit-Weighted Breu, Kirkpatrick 98 Unit-Weighted, Embedded Weighted Breu, Kirkpatrick 98 Weighted, Embedded Our results: NP-hardness statements and construction algorithms. Related work: NP-hardness statements and construction algorithms. Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Problem Variations, Results, Related Work All planar graphs are Disk Contact Graphs [Koebe, 1936] We want to have control over disk sizes and the embedding. Weighted DCG Recognition Problem: Does 1 2 3 vertex-weighted graph G have a DCR in which the disks’ radii correspond to their weights ? Embedded DCG Recognition Problem: Does b a c b plane graph G have a DCR with an according c a v rotation system ? v star caterpillar tree outerplanar planar 1. Unit-Weighted Breu, Kirkpatrick 98 Unit-Weighted, Embedded Weighted Breu, Kirkpatrick 98 Weighted, Embedded Our results: NP-hardness statements and construction algorithms. Related work: NP-hardness statements and construction algorithms. Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Problem Variations, Results, Related Work All planar graphs are Disk Contact Graphs [Koebe, 1936] We want to have control over disk sizes and the embedding. Weighted DCG Recognition Problem: Does 1 2 3 vertex-weighted graph G have a DCR in which the disks’ radii correspond to their weights ? Embedded DCG Recognition Problem: Does b a c b plane graph G have a DCR with an according c a v rotation system ? v star caterpillar tree outerplanar planar 1. Unit-Weighted Breu, Kirkpatrick 98 Bowen et al. Unit-Weighted, Embedded GD 2015 Weighted Breu, Kirkpatrick 98 Bowen et al. Weighted, Embedded GD 2015 Our results: NP-hardness statements and construction algorithms. Related work: NP-hardness statements and construction algorithms. Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Problem Variations, Results, Related Work All planar graphs are Disk Contact Graphs [Koebe, 1936] We want to have control over disk sizes and the embedding. Weighted DCG Recognition Problem: Does 1 2 3 vertex-weighted graph G have a DCR in which the disks’ radii correspond to their weights ? Embedded DCG Recognition Problem: Does b a c b plane graph G have a DCR with an according c a v rotation system ? v star caterpillar tree outerplanar planar ? 1. Unit-Weighted Breu, Kirkpatrick 98 Bowen et al. Unit-Weighted, Embedded GD 2015 Weighted Breu, Kirkpatrick 98 Bowen et al. Weighted, Embedded GD 2015 Our results: NP-hardness statements and construction algorithms. Related work: NP-hardness statements and construction algorithms. Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Problem Variations, Results, Related Work All planar graphs are Disk Contact Graphs [Koebe, 1936] We want to have control over disk sizes and the embedding. Weighted DCG Recognition Problem: Does 1 2 3 vertex-weighted graph G have a DCR in which the disks’ radii correspond to their weights ? Embedded DCG Recognition Problem: Does b a c b plane graph G have a DCR with an according c a v rotation system ? v star caterpillar tree outerplanar planar ? 2. 1. Unit-Weighted Breu, Kirkpatrick 98 Bowen et al. Unit-Weighted, Embedded GD 2015 Weighted Breu, Kirkpatrick 98 Bowen et al. Weighted, Embedded GD 2015 Our results: NP-hardness statements and construction algorithms. Related work: NP-hardness statements and construction algorithms. Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Problem Variations, Results, Related Work All planar graphs are Disk Contact Graphs [Koebe, 1936] We want to have control over disk sizes and the embedding. Weighted DCG Recognition Problem: Does 1 2 3 vertex-weighted graph G have a DCR in which the disks’ radii correspond to their weights ? Embedded DCG Recognition Problem: Does b a c b plane graph G have a DCR with an according c a v rotation system ? v star caterpillar tree outerplanar planar ? 2. 1. Unit-Weighted Breu, Kirkpatrick 98 Bowen et al. Unit-Weighted, Embedded GD 2015 3. Weighted Breu, Kirkpatrick 98 Bowen et al. Weighted, Embedded GD 2015 Our results: NP-hardness statements and construction algorithms. Related work: NP-hardness statements and construction algorithms. Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Problem Variations, Results, Related Work All planar graphs are Disk Contact Graphs [Koebe, 1936] We want to have control over disk sizes and the embedding. Weighted DCG Recognition Problem: Does 1 2 3 vertex-weighted graph G have a DCR in which the disks’ radii correspond to their weights ? Embedded DCG Recognition Problem: Does b a c b plane graph G have a DCR with an according c a v rotation system ? v star caterpillar tree outerplanar planar ? 2. 1. Unit-Weighted Breu, Kirkpatrick 98 Bowen et al. Unit-Weighted, Embedded GD 2015 3. Weighted Breu, Kirkpatrick 98 Bowen et al. 4. Weighted, Embedded GD 2015 Our results: NP-hardness statements and construction algorithms. Related work: NP-hardness statements and construction algorithms. Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Problem Variations, Results, Related Work All planar graphs are Disk Contact Graphs [Koebe, 1936] We want to have control over disk sizes and the embedding. Weighted DCG Recognition Problem: Does 1 2 3 vertex-weighted graph G have a DCR in which the disks’ radii correspond to their weights ? Embedded DCG Recognition Problem: Does b a c b plane graph G have a DCR with an according c a v rotation system ? v star caterpillar tree outerplanar planar ? 2. 1. Unit-Weighted Breu, Kirkpatrick 98 Bowen et al. Unit-Weighted, Embedded GD 2015 3. Weighted Breu, Kirkpatrick 98 Bowen et al. 4. Weighted, Embedded GD 2015 Our results: NP-hardness statements and construction algorithms. Related work: NP-hardness statements and construction algorithms. Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs

Unit-Weighted DCG Recognition Theorem: Deciding whether a graph G has a unit disk contact representation is NP-hard even if G is outerplanar . Boris Klemz, Martin N¨ ollenburg, Roman Prutkin Recognizing Weighted Disk Contact Graphs