Graph invariants | Graph minor theory
In graph theory, a path decomposition of a graph G is, informally, a representation of G as a "thickened" path graph, and the pathwidth of G is a number that measures how much the path was thickened to form G. More formally, a path-decomposition is a sequence of subsets of vertices of G such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets, and the pathwidth is one less than the size of the largest set in such a decomposition.Pathwidth is also known as interval thickness (one less than the maximum clique size in an interval supergraph of G), vertex separation number, or node searching number. Pathwidth and path-decompositions are closely analogous to treewidth and tree decompositions. They play a key role in the theory of graph minors: the families of graphs that are closed under graph minors and do not include all forests may be characterized as having bounded pathwidth, and the "vortices" appearing in the general structure theory for minor-closed graph families have bounded pathwidth. Pathwidth, and graphs of bounded pathwidth, also have applications in VLSI design, graph drawing, and computational linguistics. It is NP-hard to find the pathwidth of arbitrary graphs, or even to approximate it accurately. However, the problem is fixed-parameter tractable: testing whether a graph has pathwidth k can be solved in an amount of time that depends linearly on the size of the graph but superexponentially on k. Additionally, for several special classes of graphs, such as trees, the pathwidth may be computed in polynomial time without dependence on k.Many problems in graph algorithms may be solved efficiently on graphs of bounded pathwidth, by using dynamic programming on a path-decomposition of the graph. Path decomposition may also be used to measure the space complexity of dynamic programming algorithms on graphs of bounded treewidth. (Wikipedia).
Kristof Huszar: On the Pathwidth of Hyperbolic 3-Manifolds
Kristof Huszar, Inria Sophia Antipolis - Mediterranee, France Title: On the Pathwidth of Hyperbolic 3-Manifolds Abstract: In recent years there has been an emergence of fixed-parameter tractable (FPT) algorithms that efficiently solve hard problems for triangulated 3-manifolds as soon as t
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
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
What is the Corresponding Angle Converse Theorem
👉 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
What is an example of lines that are a linear pair
👉 Learn how to define angle relationships. Knowledge of the relationships between angles can help in determining the value of a given angle. The various angle relationships include: vertical angles, adjacent angles, complementary angles, supplementary angles, linear pairs, etc. Vertical a
From playlist Angle Relationships
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
What is an angle and it's parts
👉 Learn how to define angle relationships. Knowledge of the relationships between angles can help in determining the value of a given angle. The various angle relationships include: vertical angles, adjacent angles, complementary angles, supplementary angles, linear pairs, etc. Vertical a
From playlist Angle Relationships
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
What are acute, obtuse, right, and straight angles
👉 Learn how to define angle relationships. Knowledge of the relationships between angles can help in determining the value of a given angle. The various angle relationships include: vertical angles, adjacent angles, complementary angles, supplementary angles, linear pairs, etc. Vertical a
From playlist Angle Relationships
👉 Learn how to define angle relationships. Knowledge of the relationships between angles can help in determining the value of a given angle. The various angle relationships include: vertical angles, adjacent angles, complementary angles, supplementary angles, linear pairs, etc. Vertical a
From playlist Angle Relationships
What is the Consecutive Interior Angle Converse Theorem
👉 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
Hans Bodlaender: Parameterized Problems Complete for Nondeterministic FPT time and Logarithmic Space
Let XNLP be the class of parameterized problems such that an instance of size n with parameter k can be solved nondeterministically in time f(k)n to the power of O(1) and space f(k) log(n) (for some computable function f). We give a wide variety of XNLP-complete problems, such as List Colo
From playlist Workshop: Parametrized complexity and discrete optimization
Intersection of a Plane and a Line
Quickstart for Web and Tablet App Example 5: Intersection of a Plane and a Line
From playlist Quickstart for Web and Tablet App