48 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Graphs | Obsidian::STEM | set::graph |
|
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
Basic
Let G = \langle V, E \rangle be a directed graph. What kind of mathematical object is V?
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 kind of mathematical object is E?
Back: It is a 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 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
Basic
Let G = \langle V, E \rangle be an undirected graph. What kind of mathematical object is V?
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 kind of mathematical object is E?
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?
!
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?
!
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?
!
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 coordinate 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 coordinate 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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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 to vertex 2?
!
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?
!
Back: \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?
!
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?
!
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 in a digraph where 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}.
!
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}.
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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?
!
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 simple cycles containing vertex 2?
!
Back: \langle 2, 5, 1, 2 \rangle and \langle 2, 1, 5, 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 3?
!
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%%
Isomorphisms
An isomorphism between two graphs G_1 and G_2 is a bijection f \colon V_1 \rightarrow V_2 between the vertices of the graphs such that (a, b) is an edge in G_1 if and only if (f(a), f(b)) is an edge in G_2. Here parenthesis are used to denote either ordered pairs (for directed graphs) or unordered pairs (for undirected graphs).
We say G_1 and G_2 are isomorphic, denoted G_1 \cong G_2, if and only if there exists an isomorphism between G_1 and G_2.
%%ANKI Basic What kind of mathematical object is an isomorphism between graphs? Back: A function. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What kind of function is an isomorphism between two graphs? Back: A bijective function. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is an isomorphism between graphs G_1 = (V_1, E_1) and G_2 = (V_2, E_2)?
Back: A bijection f \colon V_1 \rightarrow V_2 such that (a, b) \in E_1 if and only if (f(a), f(b)) \in E_2.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is the domain of an isomorphism between graphs G_1 = (V_1, E_1) and G_2 = (V_2, E_2)?
Back: V_1.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is the codomain of an isomorphism between graphs G_1 = (V_1, E_1) and G_2 = (V_2, E_2)?
Back: V_2.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is the edge relation of isomorphism f between graphs G_1 = (V_1, E_1) and G_2 = (V_2, E_2)?
Back: (a, b) \in E_1 if and only if (f(a), f(b)) \in E_2.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What does it mean for graphs G_1 and G_2 to be isomorphic?
Back: There exists an isomorphism between them.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic If two graphs are equal, are they isomorphic? Back: Yes. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic If two graphs are isomorphic, are they equal? Back: Not necessarily. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Are the following two graphs equal?
!
Back: No.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Are the following two graphs isomorphic?
!
Back: Yes.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
If the following graphs are isomorphic, what is the domain of the isomorphism?
!
Back: \{a, b, c\}.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
If the following graphs are isomorphic, what is the codomain of the isomorphism?
!
Back: \{u, v, w\}.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What does it mean for two graphs to be equal? Back: Two graphs are equal if their vertex and edge sets are equal. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Cloze Graphs are to {isomorphic} as shapes are to {congruent}. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
Subgraphs
We say G' = (V', E') is a subgraph of G = (V, E) provided V' \subseteq V and E' \subseteq E. We say G' = (V', E') is an induced subgraph of G = (V, E) provided V' \subseteq V and every edge in E whose vertices are still in V' is also an edge in E'.
%%ANKI
Basic
What is a subgraph of G = (V, E)?
Back: A graph G' = (V', E') such that V' \subseteq V and E' \subseteq E.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is an induced subgraph of G = (V, E)?
Back: A graph G' = (V', E') such that V' \subseteq V and every edge in E whose vertices are in V' is in E'.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic Which of subgraphs or induced subgraphs are more general? Back: Subgraphs. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic Is an induced subgraph a subgraph? Back: Yes. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic Is a subgraph an induced subgraph? Back: Not necessarily. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic How can deletion be used to create a subgraph from a graph? Back: By deleting vertices (with connected edges) as well as any additional edges. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic How can deletion be used to create an induced subgraph from a graph? Back: By only deleting vertices and their connected edges. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Is the second graph a subgraph of the first?
!
Back: Yes.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Is the second graph an induced subgraph of the first?
!
Back: Yes.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Is the second graph a subgraph of the first?
!
Back: Yes.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Is the second graph an induced subgraph of the first?
!
Back: No.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Why isn't the second graph an induced subgraph of the first?
!
Back: The second graph is missing edge \{a, b\}.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Is the second graph a subgraph of the first?
!
Back: No.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Why isn't the second graph a subgraph of the first?
!
Back: Edge \{c, f\} is not in the first graph.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Is the second graph an induced subgraph of the first?
!
Back: No.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Why isn't the second graph an induced subgraph of the first?
!
Back: Because the second graph isn't even a subgraph of the first.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
Bibliography
- Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
- Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).