--- title: Graphs TARGET DECK: Obsidian::STEM FILE TAGS: set::graph tags: - 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 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? ![[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%% A graph that allows multiple edges between vertices is called a **multigraph**. It is analagous to the concept of [[bags|multisets]] in set theory. %%ANKI Basic What is a multigraph? Back: A graph with multiple edges between any two vertices. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Cloze {Multigraphs} are to graph theory as {multisets} are to set theory. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Does every multigraph correspond to a graph? Back: No. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Does every graph correspond to a multigraph? Back: Yes. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Under what conditions is a multigraph considered a graph? Back: When the number of edges from any vertex to any other vertex is at most $1$. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% ## Incidence 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? ![[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 {2:in}-degrees whereas incident {2: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%% ### Handshake Lemma In any graph, the sum of the degrees of vertices in the graph is always twice the number of edges: $$\sum_{v \in V} d(v) = 2e.$$ %%ANKI Basic *Why* is the handshake lemma named the way it is? Back: It invokes imagery of two vertices meeting (i.e. shaking hands). Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Does the handshake lemma apply to undirected graphs or directed graphs? Back: Both. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic In graph theory, what does the handshake lemma state? Back: For any graph, the sum of the degree of vertices is twice the number of edges. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Cloze For any graph, the {sum of the degree of vertices} is twice the {number of edges}. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic How is the handshake lemma expressed using summation notation? Back: $\sum_{v \in V} d(v) = 2e$ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Consider a graph with the following degree sequence. How many vertices are there? $$\langle 4, 4, 3, 3, 3, 2, 1 \rangle$$ Back: $7$ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Consider a graph with the following degree sequence. How many edges are there? $$\langle 4, 4, 3, 3, 3, 2, 1 \rangle$$ Back: $10$ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic *Why* is the handshake lemma true? Back: Every edge adds to the degree of two vertices. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% ## Walks Let $G = (V, E)$ be a graph. A **walk** of $G$ is a sequence of vertices such that consecutive vertices in the sequence are adjacent in $G$. More precisely, a walk (of length $k$) from vertex $v_0$ to vertex $v_k$ is a sequence $w = \langle v_0, v_1, \ldots, v_k \rangle$ of vertices such that $(v_{i-1}, v_i) \in E$ for $i = 1, 2, \ldots, k$. We say $v_k$ is **reachable** from $v_0$ via $w$. %%ANKI Basic What is a walk of (say) graph $G$? Back: A sequence of vertices such that consecutive vertices in the sequence are adjacent in $G$. 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 graph with walk $\langle v_0, v_1, \ldots, v_k \rangle$. What is the walk'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 walk? Back: The number of edges specified in the walk. 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 walk? Back: One less than the number of vertices specified in the walk. 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 walk 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 walk of a graph relate to the concept of adjacency? Back: Each vertex must be adjacent to the vertex preceding it in the underlying sequence. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic How does a walk 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 underlying sequence. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic How does a walk 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 underlying sequence. 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 walk 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 walks of length {1:$\geq 0$} whereas adjacency is to walks of length {1:$1$}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic What are the walks of length $2$ from vertex $2$ to vertex $2$? ![[directed-graph-example.png]] Back: $\langle 2, 2, 2 \rangle$ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% ### Trails A **trail** is a walk in which no edge is repeated. %%ANKI Basic What is a trail of (say) graph $G$? Back: A walk of $G$ in which no edge is repeated. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Which of walks or trails is more general? Back: Walks. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic What are the trails of length $2$ from vertex $2$ to vertex $2$? ![[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 trails of length $4$ from vertex $2$ to vertex $2$? ![[directed-graph-example.png]] Back: $\langle 2, 4, 1, 2, 2 \rangle$, $\langle 2, 2, 4, 1, 2 \rangle$, 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 What are the trails from vertex $2$ to vertex $1$? ![[undirected-graph-example.png]] Back: $\langle 2, 1 \rangle$ and $\langle 2, 5, 1 \rangle$ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% ### Paths A **path** is a trail in which no vertex is repeated (except possibly the first and last). A **cycle** is a path that starts and ends at the same vertex. A graph with no cycles is **acyclic**. In computer science, a cycle is sometimes required to have more than one edge: * In a directed graph, path $\langle v_0, v_1, \ldots, v_k \rangle$ is a cycle if $v_0 = v_k$ and the path contains at least one edge. * In an undirected graph, path $\langle v_0, v_1, \ldots, v_k \rangle$ is a cycle if $v_0 = v_k$ and all edges are distinct. %%ANKI Basic What is a path of (say) graph $G$? Back: A trail of $G$ in which no vertex is repeated (except possibly the first with the last). Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic What is a cycle of (say) graph $G$? Back: A path of $G$ that starts and ends at the same vertex. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic What is a trivial cycle of (say) graph $G$? Back: A cycle of length $0$, i.e. a single vertex. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Which of trails or paths are more general? Back: Trails. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Which of cycles or paths are more general? Back: Paths. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Which of cycles or trails are more general? Back: Trails. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). 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 $1$ 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 $4$ from vertex $2$ to vertex $2$? ![[directed-graph-example.png]] Back: $\langle 2, 5, 4, 1, 2 \rangle$ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic 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 walks 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 What are the paths from vertex $3$ to vertex $6$? ![[undirected-graph-example.png]] Back: $\langle 3, 6 \rangle$ 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 path $\langle v_0, v_1, \ldots, v_k \rangle$ considered a non-trivial 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 How many edges exist in a non-trivial 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 non-trivial 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 non-trivial 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 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 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 cycles to vertex $2$? ![[directed-graph-example.png]] Back: $\langle 2 \rangle$, $\langle 2, 2 \rangle$, $\langle 2, 4, 1, 2 \rangle$, 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 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: $\langle 4, 1, 2 \rangle$ 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: $\langle 2, 4, 1, 2 \rangle$ 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 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 to vertex $2$? ![[undirected-graph-example.png]] Back: $\langle 2 \rangle$, $\langle 2, 5, 1, 2 \rangle$ 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 to vertex $3$? ![[undirected-graph-example.png]] Back: $\langle 3 \rangle$ 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 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 **d**irected **a**cyclic **g**raph. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic Is $\langle B, D, E, J, K, B, A \rangle$ most precisely a path, trail, or walk? ![[cyclic-undirected-labelled.png]] Back: A trail. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Is $\langle B, D, E, J, K, B \rangle$ most precisely a path, trail, or walk? ![[cyclic-undirected-labelled.png]] Back: A path. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Is $\langle B, D, B, K, L \rangle$ most precisely a path, trail, or walk? ![[cyclic-undirected-labelled.png]] Back: A walk. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Is $\langle A, B, D \rangle$ most precisely a path, trail, or walk? ![[cyclic-undirected-labelled.png]] Back: A path. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). 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](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](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](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](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](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](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](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](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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Are the following two graphs equal? ![[graph-isomorphic.png]] Back: No. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Are the following two graphs isomorphic? ![[graph-isomorphic.png]] Back: Yes. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic If the following graphs are isomorphic, what is the domain of the isomorphism? ![[graph-isomorphic.png]] Back: $\{a, b, c\}$ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic If the following graphs are isomorphic, what is the codomain of the isomorphism? ![[graph-isomorphic.png]] Back: $\{u, v, w\}$ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](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](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](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](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](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](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](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](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](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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Is the second graph a subgraph of the first? ![[graph-induced-subgraph.png]] Back: Yes. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Is the second graph an induced subgraph of the first? ![[graph-induced-subgraph.png]] Back: Yes. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Is the second graph a subgraph of the first? ![[graph-subgraph.png]] Back: Yes. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Is the second graph an induced subgraph of the first? ![[graph-subgraph.png]] Back: No. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Why isn't the second graph an induced subgraph of the first? ![[graph-subgraph.png]] 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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Is the second graph a subgraph of the first? ![[graph-non-subgraph.png]] Back: No. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Why isn't the second graph a subgraph of the first? ![[graph-non-subgraph.png]] 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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Is the second graph an induced subgraph of the first? ![[graph-non-subgraph.png]] Back: No. Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). END%% %%ANKI Basic Why isn't the second graph an induced subgraph of the first? ![[graph-non-subgraph.png]] 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](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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). * Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).