notebook/notes/set/graphs.md

37 KiB

title TARGET DECK FILE TAGS tags
Graphs Obsidian::STEM set::graph
graph
set

Overview

A directed graph G is a pair \langle V, E \rangle, where V is a finite set and E is a binary relation on V. An undirected graph G is a pair \langle V, E \rangle, where V is a finite set and E is a set of unordered pair of vertices from V. In both types of graphs, V is called the vertex set of G and E is called the edge set of G.

%%ANKI Basic What two components make up a directed graph? Back: A vertex set and an edge set. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What two components make up an undirected graph? Back: A vertex set and an edge set. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What kind of graph(s) might G = \langle V, E \rangle be? Back: Directed or undirected. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze A {1:directed} graph G is a pair \langle V, E \rangle, where V is a {2:finite set} and E is a {2:binary relation on V}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be a directed graph. What conditions must V satisfy? Back: It is a finite set. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be a directed graph. What conditions must E satisfy? Back: It is a binary relation on E. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be a directed graph. What name is given to V? Back: The vertex set of G. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be a directed graph. What name is given to E? Back: The edge set of G. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be a directed graph. What name refers to the members of V? Back: Vertices. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be a directed graph. What name refers to the members of E? Back: Edges. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze An {1:undirected} graph G is a pair \langle V, E \rangle, where V is a {2:finite set} and E is a {2:set of unordered pairs of vertices}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be an undirected graph. What conditions must V satisfy? Back: It is a finite set. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be an undirected graph. What conditions must E satisfy? Back: It is a set of unordered pairs of vertices. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be an undirected graph. What name is given to V? Back: The vertex set of G. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be an undirected graph. What name is given to E? Back: The edge set of G. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be an undirected graph. What name refers to the members of V? Back: Vertices. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be an undirected graph. What name refers to the members of E? Back: Edges. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which of directed or undirected graphs allow self-loops? Back: Directed graphs. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a directed graph to be simple? Back: It has no self-loops. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the smallest change that can be made for this graph to be considered simple? !directed-graph-example.png Back: The self-loop at vertex 2 must be removed. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze A directed graph with {no self-loops} is said to be {simple}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze {1:Ordered pairs} are to {2:directed} graphs whereas {2:unordered} pairs are to {1:undirected} graphs. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a directed graph to contain a self-loop? Back: It contains an edge from a vertex to itself. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze {1:\langle u, v \rangle} is to a {2:directed} graph whereas {2:\{u, v\}} is to an {1:undirected} graph. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let \langle u, v \rangle be an edge of a directed graph. What can be said about u and v? Back: They are members of the vertex set. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let \{ u, v \} be an edge of an undirected graph. What two things can be said about u and v? Back: u \neq v and they are members of the vertex set. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why are self-loops not permitted in an undirected graph? Back: An edge \{u, v\} of an undirected graph satisfies u \neq v by definition. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How is an edge of a directed graph usually depicted pictorially? Back: As an arrow. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How is an edge of an undirected graph usually depicted pictorially? Back: As a line segment. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the following a directed or undirected graph? !directed-graph-example.png Back: Directed. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the following a directed or undirected graph? !undirected-graph-example.png Back: Undirected. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

If \langle u, v \rangle is an edge of a directed graph, we say \langle u, v \rangle is incident to v and incident from u. Furthermore, we say v is adjacent to u. If \{u, v\} was instead an edge of an undirected graph, we say \{u, v\} is incident on u and v. Likewise, v is adjacent to u and u is adjacent to v.

%%ANKI Cloze Let \langle u, v \rangle be an edge of a directed graph. Then {1:\langle u, v \rangle} is incident from {1:u}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze Let \langle u, v \rangle be an edge of a directed graph. Then {1:\langle u, v \rangle} is incident to {1:v}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for an edge to be incident from vertex v? Back: v is the first member of the edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for an edge to be incident to vertex v? Back: v is the second member of the edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic With respect to directed graphs, what term describes an edge of form \langle v, v \rangle? Back: A self-loop. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which edges are incident from vertex 2 in the following? !directed-graph-example.png Back: \langle 2, 2 \rangle, \langle 2, 4 \rangle, \langle 2, 5 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which edges are incident to vertex 2 in the following? !directed-graph-example.png Back: \langle 1, 2 \rangle, \langle 2, 2 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What "kinds" of incidence exist in a directed graph? Back: Incidence to and incidence from. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Given directed graph G = \langle V, E \rangle, what does it mean for vertex u to be adjacent to v? Back: There exists an edge \langle v, u \rangle in E. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Given directed graph G = \langle V, E \rangle, what does it mean for vertex v to be adjacent to u? Back: There exists an edge \langle u, v \rangle in E. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Given undirected graph G = \langle V, E \rangle, what does it mean for vertex v to be adjacent to u? Back: There exists an edge \{ u, v \} in E. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze Let \langle u, v \rangle be an edge of an undirected graph. Then {1:\langle u, v \rangle} is incident on {1:u and v}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for an edge to be incident on vertex v? Back: v is a member of the edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze Incident {1:to/from} is to directed graphs whereas incident {1:on} is to undirected graphs. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which edges are incident on vertex 2 in the following? !undirected-graph-example.png Back: \{ 1, 2 \}, \{2, 5\} Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What "kinds" of incidence exist in an undirected graph? Back: Incidence on. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the concept of adjacency related to directed graphs or undirected graphs? Back: Both. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the concept of incidence related to directed graphs or undirected graphs? Back: Both. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Adjacency is a binary relation on what two kinds of objects? Back: Vertices. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a directed graph, how can we restate "vertex v is adjacent to vertex u" in terms of incidence to? Back: Edge \langle u, v \rangle is incident to v. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a directed graph, how can we restate "vertex v is adjacent to vertex u" in terms of incidence from? Back: Edge \langle u, v \rangle is incident from u. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a directed graph, how can we restate "edge \langle u, v \rangle is incident to v" in terms of adjacency? Back: Vertex v is adjacent to vertex u. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a directed graph, how can we restate "edge \langle u, v \rangle is incident from u" in terms of adjacency? Back: Vertex v is adjacent to vertex u. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Incidence is a binary relation on what two kinds of objects? Back: A vertex and an edge. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In an undirected graph, how can we restate "vertex v is adjacent to vertex u" in terms of incidence on? Back: Edge \{u, v\} is incident on v. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In an undirected graph, how can we restate "vertex u is adjacent to vertex v" in terms of incidence on? Back: Edge \{v, u\} is incident on u. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In what kind of graph is adjacency necessarily symmetric? Back: Undirected graphs. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In what kind of graph is adjacency not necessarily symmetric? Back: Directed graphs. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which vertices is vertex 2 adjacent to? !directed-graph-example.png Back: 1 and 2. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which vertices is vertex 2 adjacent to? !undirected-graph-example.png Back: 1 and 5. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the degree of a vertex of a directed graph? Back: The number of edges incident to and from the vertex. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a directed graph, how is a vertex's degree further subcategorized? Back: As in-degrees and out-degrees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the in-degree of a vertex of a directed graph? Back: The number of edges incident to the vertex. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the out-degree of a vertex of a directed graph? Back: The number of edges incident from the vertex. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze Given a directed graph, incident {1:to} is to {1:in}-degrees whereas incident {1:from} is to {1:out}-degrees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the in-degree of vertex 5? !directed-graph-example.png Back: 2. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the out-degree of vertex 5? !directed-graph-example.png Back: 1. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the degree of vertex 4? !directed-graph-example.png Back: 4 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the degree of a vertex of an undirected graph? Back: The number of edges incident on the vertex. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the degree of vertex 3? !undirected-graph-example.png Back: 1 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a vertex of a graph to be isolated? Back: It has degree 0. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What term describes a vertex of a graph with degree 0? Back: Isolated. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which vertices are isolated in the following? !directed-graph-example.png Back: N/A Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which vertices are isolated in the following? !undirected-graph-example.png Back: 4 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What term describes vertex 4 in the following? !undirected-graph-example.png Back: Isolated. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Paths

A path of length k from a vertex u to vertex u' is a sequence p = \langle v_0, v_1, \ldots, v_k \rangle of vertices such that u = v_0, u' = v_k, and (v_{i-1}, v_i) \in E for i = 1, 2, \ldots, k. In this case, we say u' is reachable from u via p. A path is simple if all vertices in the path are distinct.

%%ANKI Basic Let G = \langle V, E \rangle be a graph. What is a path from vertex u to vertex v? Back: A sequence of vertices \langle u, \ldots, v \rangle such that there is an edge for each consecutive pair of vertices. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be a graph with path \langle v_0, v_1, \ldots, v_k \rangle. What is the path's length? Back: k Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of edges, what is the length of a path? Back: The number of edges specified in the path. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of vertices, what is the length of a path? Back: One less than the number of vertices specified in the path. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let G = \langle V, E \rangle be a graph. A path of G is said to contain what? Back: Vertices and edges. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How does a path of a graph relate to the concept of adjacency? Back: Each vertex must be adjacent to the vertex preceding it in the path. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How does a path of a directed graph relate to the concept of incidence? Back: There exists an edge incident to each vertex that is also incident from the vertex preceding it in the path. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How does a path of an undirected graph relate to the concept of incidence? Back: There exists an edge incident on each vertex and the vertex preceding it in the path. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths from vertex 3 to vertex 6? !directed-graph-example.png Back: N/A Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths from vertex 6 to vertex 3? !directed-graph-example.png Back: \langle 6, 3 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths from vertex 6 to vertex 6? !directed-graph-example.png Back: \langle 6 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths of length 1 from vertex 2 to vertex 2? !directed-graph-example.png Back: \langle 1, 2 \rangle, \langle 2, 2 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths of length 2 from vertex 2 to vertex 2? !directed-graph-example.png Back: \langle 4, 1, 2 \rangle, \langle 2, 2, 2 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths from vertex 4 to vertex 4? !undirected-graph-example.png Back: \langle 4 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths from vertex 3 to vertex 6? !undirected-graph-example.png Back: \langle 3, 6 \rangle, \langle 3, 6, 3, 6 \rangle, \ldots Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Reachability is a binary relation on what two kinds of objects? Back: Vertices. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How does reachability relate to adjacency? Back: Reachability is the transitive generalization of adjacency. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What proximity-based term describes distinct vertices being maximally close? Back: Adjacency. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze {Reachability} is the generalization of {adjacency}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for vertex u to be reachable to vertex v? Back: There exists a path from u to v. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What path must exist if vertex u is adjacent to vertex v? Back: \langle v, u \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze Reachable is to paths of length {1:\geq 0} whereas adjacency is to paths of length {1:1}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

A path is simple if all vertices in the path are distinct. In a directed graph, path \langle v_0, v_1, \ldots, v_k \rangle forms a cycle if v_0 = v_k and the path contains at least one edge. In an undirected graph, path \langle v_0, v_1, \ldots, v_k \rangle forms a cycle if v_0 = v_k and all edges are distinct. We say a cycle is simple if all vertices in the path (barring the first and last) are distinct. A graph with no simple cycles is acyclic.

%%ANKI Basic What does it mean for a path to be simple? Back: All the vertices in the path are distinct. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a directed graph, when is \langle v_0, v_1, \ldots, v_k \rangle considered a cycle? Back: When v_0 = v_k and there is at least one edge in the path. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of edges, what is the length of a cycle? Back: The number of edges specified in the path. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In terms of vertices, what is the length of a cycle? Back: One less than the number of vertices specified in the path. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a cycle to be simple? Back: Except for the first which equals the last, all the vertices in the path are distinct. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many edges exist in a cycle of a directed graph? Back: At least one. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In an undirected graph, when is \langle v_0, v_1, \ldots, v_k \rangle considered a cycle? Back: When v_0 = v_k, k > 0, and all edges in the path are distinct. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many edges exist in a cycle of an undirected graph? Back: At least three. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze Path \langle 1, 2, 4, 1 \rangle is not a simple {1:path} but is a simple {1:cycle}. !directed-graph-example.png Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze Path \langle 1, 2, 4 \rangle is a simple {1:path} but not a simple {1:cycle}. !directed-graph-example.png Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic With respect to paths, what ambiguity exists with the term "simple"? Back: Whether we are referring to simple paths or simple cycles. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths to vertex 3? !directed-graph-example.png Back: \langle 3 \rangle and \langle 6, 3 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths to vertex 6? !directed-graph-example.png Back: \langle 6 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the simple paths of length 1 to vertex 2? !directed-graph-example.png Back: \langle 1, 2 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths of length 1 to vertex 2? !directed-graph-example.png Back: \langle 1, 2 \rangle and \langle 2, 2 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the paths of length 2 to vertex 2? !directed-graph-example.png Back: \langle 4, 1, 2 \rangle and \langle 2, 2, 2 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the cycles of length 3 to vertex 2? !directed-graph-example.png Back: \langle 2, 4, 1, 2 \rangle and \langle 2, 2, 2, 2 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the simple cycles of length 3 to vertex 2? !directed-graph-example.png Back: \langle 2, 4, 1, 2 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are all the simple cycles containing vertex 2? !directed-graph-example.png Back: \langle 2, 2 \rangle, \langle 2, 4, 1, 2 \rangle, and \langle 2, 5, 4, 1, 2 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why isn't \langle 3, 6, 3 \rangle considered a cycle? !undirected-graph-example.png Back: All the edges in the path must be distinct. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why isn't \langle 3, 6 \rangle considered a cycle? !undirected-graph-example.png Back: The first and last vertex of the path must be the same. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the simple paths to vertex 2? !undirected-graph-example.png Back: \langle 2 \rangle, \langle 1, 2 \rangle, \langle 5, 2 \rangle, \langle 1, 5, 2 \rangle, \langle 5, 1, 2 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the cycles containing vertex 2? !undirected-graph-example.png Back: \langle 1, 2, 5, 1 \rangle and \langle 1, 5, 2, 1 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the simple cycles containing vertex 2? !undirected-graph-example.png Back: \langle 1, 2, 5, 1 \rangle and \langle 1, 5, 2, 1 \rangle Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the cycles containing vertex 3? !undirected-graph-example.png Back: N/A Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a graph to be acyclic? Back: It has no simple cycles. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is DAG an acronym for? Back: A directed acyclic graph. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

References

  • Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).