PPT-Planar Graphs & Euler’s Formula

Exercise 82 Isomorphic Graphs Graphs with same number of vertices edges amp the same number of connections between vertices If vertices labelled same way matrix representation will be the same. Carla . Binucci. , Emilio Di Giacomo, . Walter Didimo, Fabrizio Montecchiani, Maurizio . Patrignani. , . Ioannis. G. . Tollis. Fan-planar drawings. Fan-planar drawings. Given a graph G, a . fan-planar drawing . Dwyane George. March 10, 2015. Outline. Definitions. Algorithm: Single-Source Shortest Path (SSSP. ). r. -Division (Clustering). Runtime Analysis. Proof . of . Correctness. Planar Graphs. Definition: a graph G that can be embedded in a plane, such that the edges only intersect at the endpoints. Angelika Steger. (j. oint. . work. . with. . Konstantinos . Panagiotou. , SODA‘11. ) . . TexPoint fonts used in EMF. . Read the TexPoint manual before you delete this box.: . A. A. A. A. A. Random Graphs . Graphs. Fall . 2011. Sukumar Ghosh. Seven Bridges of . K. ⍥. nigsberg. Is it possible to walk along a route that cross . each bridge exactly once?. Seven Bridges of . K. ⍥. nigsberg. A Graph. What is a Graph. Lecture 20: Nov 25. This Lecture. Graph coloring is another important problem in graph theory.. It also has many applications, including the famous 4-color problem.. Graph coloring. Applications. Planar graphs. When an Euler path is impossible, we can get an approximate path. In the approximate path, some edges will need to be retraced. An . optimal approximation. of a Euler path is a path with the minimum number of edge . Minors, . Bidimensionality. ,. & Decomposition. r. r. Erik Demaine. MIT. Goals. How far . beyond planar graphs . can we go?. Graphs excluding. a fixed minor. Powers thereof. Build . general approximation frameworks . . Intro problem- 3 houses and 3 utilities.  K. 3,3. problem: Can 3 houses be connected to 3 utilities so that no 2 lines cross?. Similarly, can an isomorphic version of K. 3,3. be drawn in the plane so that no two edges cross?. on Cops and Robbers. Anthony Bonato. Ryerson University. Genus. (Aigner, . Fromme. , 84) . planar graphs (genus . 0. ) have cop number . ≤ 3.. (Clarke, 02) . outerplanar. graphs have cop number . Eyal. Ackerman. University of Haifa and . Oranim. College. Drawing graphs in the plane. Consider drawings of graphs in the plane . s.t. .. No loops or parallel edges. Vertices .  distinct points. Daniel Lokshtanov. Based on joint work with Hans Bodlaender ,Fedor Fomin,Eelko Penninkx, Venkatesh Raman, Saket Saurabh and Dimitrios Thilikos. Background. Most interesting graph problems are . NP-hard. Anthony Bonato. Ryerson University. GRASCan’17. Grenfell Campus. Graphs on surfaces. ?. ?. S. 0. S. 1. Genus of a graph. a graph that can be embedded in an (orientable!) surface with . g. holes (and . Planar graphs. 2. Planar graphs. Can be drawn on the plane without crossings. Plane graph: planar graph, given together with an embedding in the plane. Many applications…. Questions:. Testing if a graph is planar. Chapter 6: Graphs 6.2 The Euler Characteristic Draw A Graph! Any connected graph you want, but don’t make it too simple or too crazy complicated Only rule: No edges can cross (unless there’s a vertex where they’re crossing)