Topological graph theory | Graph operations | Duality theories | Planar graphs | Algebraic graph theory
In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized combinatorially by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces. These notions of dual graphs should not be confused with a different notion, the edge-to-vertex dual or line graph of a graph. The term dual is used because the property of being a dual graph is symmetric, meaning that if H is a dual of a connected graph G, then G is a dual of H. When discussing the dual of a graph G, the graph G itself may be referred to as the "primal graph". Many other graph properties and structures may be translated into other natural properties and structures of the dual. For instance, cycles are dual to cuts, spanning trees are dual to the complements of spanning trees, and simple graphs (without parallel edges or self-loops) are dual to 3-edge-connected graphs. Graph duality can help explain the structure of mazes and of drainage basins. Dual graphs have also been applied in computer vision, computational geometry, mesh generation, and the design of integrated circuits. (Wikipedia).
Parallel Edges in Multigraphs and Digraphs | Graph Theory, Multiple Edges, Multisets
What are parallel edges, also called multiple edges or multi-edges, in graph theory? We'll introduce parallel edges in the context of undirected multi-graphs and in directed graphs in today's video graph theory lesson! Lesson on directed graphs: https://www.youtube.com/watch?v=mXoiHgH4mE
From playlist Graph Theory
What are Connected Graphs? | Graph Theory
What is a connected graph in graph theory? That is the subject of today's math lesson! A connected graph is a graph in which every pair of vertices is connected, which means there exists a path in the graph with those vertices as endpoints. We can think of it this way: if, by traveling acr
From playlist Graph Theory
Introduction and overview of multigraphs in graph theory
From playlist Graph Theory
What are parallel lines and a transversal
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal
Graph Theory: 09. Graph Isomorphisms
In this video I provide the definition of what it means for two graphs to be isomorphic. I illustrate this with two isomorphic graphs by giving an isomorphism between them, and conclude by discussing what it means for a mapping to be a bijection. An introduction to Graph Theory by Dr. Sar
From playlist Graph Theory part-2
Dual basis definition and proof that it's a basis In this video, given a basis beta of a vector space V, I define the dual basis beta* of V*, and show that it's indeed a basis. We'll see many more applications of this concept later on, but this video already shows that it's straightforwar
From playlist Dual Spaces
What are the Angle Relationships for Parallel Lines and a Transversal
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal
This video defines and gives and example of isomorphic graphs. mathispower4u.com
From playlist Graph Theory (Discrete Math)
Proving Parallel Lines with Angle Relationships
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal
Euler's Formula and Graph Duality
A description of planar graph duality, and how it can be applied in a particularly elegant proof of Euler's Characteristic Formula. Music: Wyoming 307 by Time For Three
From playlist 3Blue1Brown | Math for fun and glory | Khan Academy
Jeff Erickson - Lecture 3 - Two-dimensional computational topology - 20/06/18
School on Low-Dimensional Geometry and Topology: Discrete and Algorithmic Aspects (http://geomschool2018.univ-mlv.fr/) Jeff Erickson (University of Illinois at Urbana-Champaign, USA) Two-dimensional computational topology - Lecture 3 Abstract: This series of lectures will describe recent
From playlist Jeff Erickson - School on Low-Dimensional Geometry and Topology: Discrete and Algorithmic Aspects
Daniel Kral: Parametrized approach to block structured integer programs
Integer programming is one of the most fundamental problems in discrete optimization. While integer programming is computationally hard in general, there exist efficient algorithms for special instances. In particular, integer programming is fixed parameter tractable when parameterized by
From playlist Workshop: Parametrized complexity and discrete optimization
Jeff Erickson - Lecture 4 - Two-dimensional computational topology - 21/06/18
School on Low-Dimensional Geometry and Topology: Discrete and Algorithmic Aspects (http://geomschool2018.univ-mlv.fr/) Jeff Erickson (University of Illinois at Urbana-Champaign, USA) Two-dimensional computational topology - Lecture 4 Abstract: This series of lectures will describe recent
From playlist Jeff Erickson - School on Low-Dimensional Geometry and Topology: Discrete and Algorithmic Aspects
Rico Zenklusen, Vera Traub: Bridging the Gap Between Tree and Connectivity Augmentation
Full title: Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches The Connectivity Augmentation Problem (CAP) is one of the most basic survivable network design problems. The task is to increase the edge-connectivity of a graph G by one unit by adding
From playlist Workshop: Continuous approaches to discrete optimization
Dimers and circle patterns by Sanjay Ramassamy
PROGRAM :UNIVERSALITY IN RANDOM STRUCTURES: INTERFACES, MATRICES, SANDPILES ORGANIZERS :Arvind Ayyer, Riddhipratim Basu and Manjunath Krishnapur DATE & TIME :14 January 2019 to 08 February 2019 VENUE :Madhava Lecture Hall, ICTS, Bangalore The primary focus of this program will be on the
From playlist Universality in random structures: Interfaces, Matrices, Sandpiles - 2019
Tao Hou (5/13/20): Computing minimal persistent cycles: Polynomial and hard cases
Title: Computing minimal persistent cycles: Polynomial and hard cases Abstract: Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in the purely topological persistence diagrams (also termed as barcodes). In our ear
From playlist AATRN 2020
Geoffrey Grimmett (University of Cambridge, UK) by Geoffrey Grimmett
PROGRAM FIRST-PASSAGE PERCOLATION AND RELATED MODELS (HYBRID) ORGANIZERS: Riddhipratim Basu (ICTS-TIFR, India), Jack Hanson (City University of New York, US) and Arjun Krishnan (University of Rochester, US) DATE: 11 July 2022 to 29 July 2022 VENUE: Ramanujan Lecture Hall and online This
From playlist First-Passage Percolation and Related Models 2022 Edited
Advanced Knowledge Problem of the Week: 4-6-17
Ryan graphs!
From playlist Center of Math: Problems of the Week
Algebraic curves, tropical geometry, and moduli - Sam Payne
Sam Payne Yale University February 11, 2015 Tropical geometry gives a new approach to understanding old questions about algebraic curves and their moduli spaces, synthesizing techniques that range from Berkovich spaces to elementary combinatorics. I will discuss an outline of this method,
From playlist Mathematics
From playlist 3d graphs