CS762: Graph-Theoretic Algorithms Felix Zhou Spring 2020, University of Waterloo from Professor Therese Biedl’s Lectures 2 Contents 1 Introduction 13 1.1 Motivation . 13 1.1.1 Points of Inquisition . 13 1.2 Graph Assumptions . 13 1.3 Weighted Dominating Set in Paths . 14 I Planar Graphs 15 2 Planar Graphs 17 2.1 Definitions . 17 2.1.1 Planar Drawing . 17 2.1.2 Planar Embedding . 18 2.1.3 Multiple Planar Embeddings . 19 2.2 Exploting the Planar Embedding . 19 2.2.1 Right-First Search . 19 2.2.2 The Menger Problem in st-planar graphs . 21 Pseudocode . 21 Analysis . 21 2.3 Dual Graphs . 21 2.3.1 Computing & Storing G∗ ........................ 22 2.3.2 Algorithmic Implications . 23 3 2.4 Euler’s Formula . 23 2.4.1 Algorithmic Implications . 24 Colouring . 25 Testing Adjacency . 25 Clique . 25 3 Problems in Planar Graphs 27 3.1 NP-Hard Problems in Planar Graphs . 27 3.1.1 Coloring . 27 3-Coloring Reduction . 28 3.1.2 Planar 3-SAT . 28 3.1.3 Independent Set . 29 Planar Reduction . 30 3.2 Maximum-Flow . 31 3.2.1 st-Cuts . 31 3.2.2 Undirected Flow in st-Planar Graphs . 32 Restricting to st-Planar Graphs . 32 4 Planarity Testing 37 4.1 Bush Forms . 37 4.2 The Algorithm by Haeupler & Tarjan . 38 4.2.1 Depth-First Search & Bush Forms . 38 4.2.2 High-Level Idea . 40 4.2.3 PQ-Trees . 40 Reductions . 41 4.2.4 Data Structures . 41 Descendants Consisting of Finished Children . 42 The Active Child . 42 4 4.2.5 Summary . 42 Initialization . 42 Tree Edge Update . 43 None-Tree Edge Update . 43 Tree Edge Retreat Update . 43 4.2.6 Final Thoughts . 44 5 Triangulated Graphs 45 5.1 Maximal Planar Graphs . 45 5.2 Related Graph Classes . 46 5.3 Canonical Ordering . 47 5.3.1 Properties . 48 5.3.2 Existence of the Canonical Order . 49 5.3.3 Splitting into Trees . 50 Arboricity . 51 5.3.4 Visibility Representation . 51 5.3.5 Straight-Line Embeddings . 53 6 Friends of Planar Graphs 55 6.1 Super Classes of Planar Graphs . 55 6.1.1 Graphs in 3D . 55 6.1.2 Graphs of Bounded Genus . 56 6.1.3 Near Planar Graphs . 56 6.2 Subclasses of Planar Graphs . 57 6.2.1 Trees . 57 6.2.2 Outer-Planar Graphs . 57 Maximal Outer-Planar Graphs . 58 Problems . 59 5 6.2.3 k-Outer-Planar Graphs . 59 6.2.4 Series-Parallel Graphs . 60 6.2.5 2-Terminal Series-Parallel Graphs . 60 The SP-Tree . 60 SP-Graphs . 61 6.2.6 Apollonion Networks . 61 6.2.7 Relationships between Subclasses of Planar Graphs . 62 II From Interval Graphs to Treewidth 63 7 Interval Graphs & Friends 65 7.1 Interval Graphs . 65 7.2 Chordal Graphs . 66 7.3 Perfect Elimination Order . 66 7.4 Problems in Chordal Graphs . 67 7.4.1 Coloring . 67 7.4.2 Clique . 68 7.4.3 Independent Set . 68 7.4.4 Dominating Set . 68 7.5 Friends of Interval Graphs . 69 7.5.1 Intersection Graphs . 69 7.5.2 H-Free Graphs . 70 7.5.3 Perfect Graphs . 70 8 Recognizing Chordal Graphs & Interval Graphs 71 8.1 Finding a Perfect Elimination Order . 71 8.1.1 Finding Simplicial Vertices . 71 8.1.2 Maximum Cardinality Search . 72 6 8.1.3 Lexicographic BFS . 74 8.2 Testing a Putative Perfect Elmination Order . 74 8.2.1 An Idea . 74 8.3 Recognizing Interval Graphs . 75 8.3.1 PQ-Trees . 78 8.3.2 Other Algorithms . 79 9 Tree Decompositions 81 9.1 Strong Path Decomposition . 81 9.2 Strong Tree Decomposition . 81 9.2.1 Perfect Elimination Orders & Tree Decompositions . 83 10 Treewidth 87 10.1 k-Trees . 87 10.1.1 Properties of k-Trees . 88 10.1.2 Planar k-Trees . 88 10.2 Partial k-Trees . 89 10.3 Treewidth . 91 10.3.1 Properties of the Treewidth . 92 10.3.2 Graphs with Big Treewidth . 93 10.3.3 Series-Parallel Graphs . 93 SP-Graphs & Treewidth . 94 Recognizing SP-Graphs . 94 11 Branchwidth 97 11.1 e-Separations & Branch Decompositions . 97 11.2 Branchwidth & Treewidth . 99 11.3 Branch Decomposition of Planar Graphs . 100 11.3.1 Spanning Trees of Small Height . 101 7 12 Pathwidth 105 12.1 Path Decomposition . ..