Edge Guarding Plane Graphs March 17, 2020 Paul Jungeblut, Torsten - PowerPoint PPT Presentation

Edge Guarding Plane Graphs March 17, 2020 Paul Jungeblut, Torsten Ueckerdt I NSTITUTE OF T HEORETICAL I NFORMATICS · A LGORITHMICS G ROUP KIT – The Research University in the Helmholtz Association www.kit.edu

Edge Guarding G = ( V , E ) plane graph. vw guards face f if at least one from { v , w } is on the boundary of f. 1 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Edge Guarding G = ( V , E ) plane graph. vw guards face f if at least one from { v , w } is on the boundary of f. 1 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Edge Guarding G = ( V , E ) plane graph. vw guards face f if at least one from { v , w } is on the boundary of f. 1 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Edge Guarding G = ( V , E ) plane graph. vw guards face f if at least one from { v , w } is on the boundary of f. 1 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Edge Guarding G = ( V , E ) plane graph. vw guards face f if at least one from { v , w } is on the boundary of f. Question For all n -vertex graphs of a planar graph class C : How many guards are sometimes necessary and always sufficient? 1 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Previous Results Lower Upper � n �� 3 n � n � 1 �� 2 � 3 + α Planar min , 3 8 9 � 4 n − 8 � n � 1 � 3 Triangulation 13 3 � n � n � 1 � 4 Outerplanar 3 3 � n � n � 5 � 5 Max. Outerplanar 4 4 α : number of quadrilateral faces 1 Bose, Shermer, Toussaint, Zhu 1997 2 Biniaz, Bose, Ooms, Verdonschot 2019 3 Everett, Rivera-Campo 1997 4 Chv´ atal 1975 5 O’Rourke 1983 2 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Our Results Lower Upper � 2 n − 4 � 2 n � � Stacked Triangulations 7 7 � n − 2 � n � � Quadrangulations 4 3 � n − 2 � n � � 2-Degenerate Quadrangulations 4 4 3 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Our Results Lower Upper � 2 n − 4 � 2 n � � Stacked Triangulations Today! 7 7 � n − 2 � n � � Quadrangulations 4 3 � n − 2 � n � � 2-Degenerate Quadrangulations 4 4 3 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation. 4 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation. 4 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation. 4 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation. 4 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation. 4 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation. Theorem [J. 2019] � 2 n � For n -vertex stacked triangulations edge guards are 7 always sufficient. 4 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion Use induction on the number n of vertices: 1. Create smaller graph G ′ of size | G ′ | = | G | − k . 2. Apply induction hypothesis on G ′ to get edge guard set Γ ′ . 3. Reinsert old vertices. 4. Use ℓ additional edges to augment Γ ′ into Γ for G . 5 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion Use induction on the number n of vertices: 1. Create smaller graph G ′ of size | G ′ | = | G | − k . 2. Apply induction hypothesis on G ′ to get edge guard set Γ ′ . 3. Reinsert old vertices. 4. Use ℓ additional edges to augment Γ ′ into Γ for G . 5 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion Use induction on the number n of vertices: 1. Create smaller graph G ′ of size | G ′ | = | G | − k . 2. Apply induction hypothesis on G ′ to get edge guard set Γ ′ . 3. Reinsert old vertices. 4. Use ℓ additional edges to augment Γ ′ into Γ for G . 5 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion Use induction on the number n of vertices: 1. Create smaller graph G ′ of size | G ′ | = | G | − k . 2. Apply induction hypothesis on G ′ to get edge guard set Γ ′ . 3. Reinsert old vertices. 4. Use ℓ additional edges to augment Γ ′ into Γ for G . 5 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion Use induction on the number n of vertices: 1. Create smaller graph G ′ of size | G ′ | = | G | − k . 2. Apply induction hypothesis on G ′ to get edge guard set Γ ′ . 3. Reinsert old vertices. 4. Use ℓ additional edges to augment Γ ′ into Γ for G . 5 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion Use induction on the number n of vertices: 1. Create smaller graph G ′ of size | G ′ | = | G | − k . 2. Apply induction hypothesis on G ′ to get edge guard set Γ ′ . 3. Reinsert old vertices. 4. Use ℓ additional edges to augment Γ ′ into Γ for G . � 2 n k ≤ 2 ℓ � 7 in all cases ⇒ edge guard set of size 7 5 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion Use induction on the number n of vertices: 1. Create smaller graph G ′ of size | G ′ | = | G | − k . 2. Apply induction hypothesis on G ′ to get edge guard set Γ ′ . 3. Reinsert old vertices. 4. Use ℓ additional edges to augment Γ ′ into Γ for G . � 2 n k ≤ 2 ℓ � 7 in all cases ⇒ edge guard set of size 7 � ℓ k ≤ 1 � Also applied successfully for 2-Degenerate Quadrangulations . 4 5 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples stacked triangulation 6 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples stacked triangulation Remove inner vertices ( k = 6). 6 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples stacked triangulation Remove inner vertices ( k = 6). Apply induction. 6 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples stacked triangulation Remove inner vertices ( k = 6). Apply induction. Reinsert inner vertices. 6 Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group