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@ -655,21 +655,88 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
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<!--ID: 1710807788304-->
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END%%
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## Paths
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### Handshake Lemma
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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.
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In any graph, the sum of the degrees of vertices in the graph is always twice the number of edges: $$\sum_{v \in V} d(v) = 2e.$$
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%%ANKI
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Basic
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Let $G = \langle V, E \rangle$ be a graph. What *is* a path from vertex $u$ to vertex $v$?
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Back: A sequence of vertices $\langle u, \ldots, v \rangle$ such that there is an edge for each consecutive pair of vertices.
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*Why* is the handshake lemma named the way it is?
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Back: It invokes imagery of two vertices meeting (i.e. shaking hands).
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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).
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<!--ID: 1723992099102-->
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END%%
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%%ANKI
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Basic
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Does the handshake lemma apply to undirected graphs or directed graphs?
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Back: Both.
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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).
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<!--ID: 1723992099108-->
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END%%
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%%ANKI
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Basic
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In graph theory, what does the handshake lemma state?
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Back: For any graph, the sum of the degree of vertices is twice the number of edges.
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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).
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<!--ID: 1723992099111-->
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END%%
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%%ANKI
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Cloze
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For any graph, the {sum of the degree of vertices} is twice the {number of edges}.
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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).
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<!--ID: 1723992099116-->
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END%%
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%%ANKI
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Basic
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How is the handshake lemma expressed using summation notation?
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Back: $\sum_{v \in V} d(v) = 2e$
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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).
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<!--ID: 1723992099120-->
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END%%
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%%ANKI
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Basic
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Consider a graph with the following degree sequence. How many vertices are there? $$\langle 4, 4, 3, 3, 3, 2, 1 \rangle$$
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Back: $7$
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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).
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<!--ID: 1723992099125-->
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END%%
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%%ANKI
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Basic
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Consider a graph with the following degree sequence. How many edges are there? $$\langle 4, 4, 3, 3, 3, 2, 1 \rangle$$
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Back: $10$
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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).
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<!--ID: 1723992099129-->
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END%%
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%%ANKI
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Basic
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*Why* is the handshake lemma true?
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Back: Every edge adds to the degree of two vertices.
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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).
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<!--ID: 1723992099134-->
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END%%
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## Walks
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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 $w$.
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%%ANKI
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Basic
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What is a walk of (say) graph $G$?
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Back: A sequence of vertices such that consecutive vertices in the sequence are adjacent in $G$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788307-->
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END%%
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%%ANKI
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Basic
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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?
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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?
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Back: $k$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788310-->
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@ -677,23 +744,23 @@ END%%
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%%ANKI
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Basic
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In terms of edges, what is the length of a path?
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Back: The number of edges specified in the path.
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In terms of edges, what is the length of a walk?
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Back: The number of edges specified in the walk.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788313-->
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END%%
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%%ANKI
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Basic
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In terms of vertices, what is the length of a path?
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Back: One less than the number of vertices specified in the path.
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In terms of vertices, what is the length of a walk?
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Back: One less than the number of vertices specified in the walk.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788317-->
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END%%
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%%ANKI
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Basic
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Let $G = \langle V, E \rangle$ be a graph. A path of $G$ is said to contain what?
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Let $G = \langle V, E \rangle$ be a graph. A walk of $G$ is said to contain what?
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Back: Vertices and edges.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788320-->
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@ -701,28 +768,195 @@ END%%
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%%ANKI
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Basic
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How does a path of a graph relate to the concept of adjacency?
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Back: Each vertex must be adjacent to the vertex preceding it in the path.
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How does a walk of a graph relate to the concept of adjacency?
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Back: Each vertex must be adjacent to the vertex preceding it in the underlying sequence.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788323-->
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END%%
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%%ANKI
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Basic
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How does a path of a directed graph relate to the concept of incidence?
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Back: There exists an edge incident to each vertex that is also incident from the vertex preceding it in the path.
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How does a walk of a directed graph relate to the concept of incidence?
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Back: There exists an edge incident to each vertex that is also incident from the vertex preceding it in the underlying sequence.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788326-->
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END%%
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%%ANKI
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Basic
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How does a path of an undirected graph relate to the concept of incidence?
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Back: There exists an edge incident on each vertex and the vertex preceding it in the path.
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How does a walk of an undirected graph relate to the concept of incidence?
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Back: There exists an edge incident on each vertex and the vertex preceding it in the underlying sequence.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788329-->
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END%%
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%%ANKI
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Basic
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Reachability is a binary relation on what two kinds of objects?
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Back: Vertices.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788359-->
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END%%
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%%ANKI
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Basic
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How does reachability relate to adjacency?
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Back: Reachability is the transitive generalization of adjacency.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788364-->
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END%%
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%%ANKI
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Basic
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What proximity-based term describes distinct vertices being maximally close?
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Back: Adjacency.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788370-->
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END%%
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%%ANKI
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Cloze
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{Reachability} is the generalization of {adjacency}.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788375-->
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END%%
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%%ANKI
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Basic
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What does it mean for vertex $u$ to be reachable to vertex $v$?
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Back: There exists a walk from $u$ to $v$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788379-->
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END%%
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%%ANKI
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Basic
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What path must exist in a digraph where vertex $u$ is adjacent to vertex $v$?
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Back: $\langle v, u \rangle$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788383-->
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END%%
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%%ANKI
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Cloze
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Reachable is to walks of length {1:$\geq 0$} whereas adjacency is to walks of length {1:$1$}.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788388-->
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END%%
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%%ANKI
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Basic
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What are the walks of length $2$ from vertex $2$ to vertex $2$?
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![[directed-graph-example.png]]
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Back: $\langle 2, 2, 2 \rangle$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788348-->
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END%%
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### Trails
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A **trail** is a walk in which no edge is repeated.
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%%ANKI
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Basic
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What is a trail of (say) graph $G$?
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Back: A walk of $G$ in which no edge is repeated.
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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).
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<!--ID: 1723992099138-->
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END%%
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%%ANKI
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Basic
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Which of walks or trails is more general?
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Back: Walks.
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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).
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<!--ID: 1723992099148-->
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END%%
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%%ANKI
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Basic
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What are the trails of length $2$ from vertex $2$ to vertex $2$?
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![[directed-graph-example.png]]
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Back: N/A.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723992099157-->
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END%%
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%%ANKI
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Basic
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What are the trails of length $4$ from vertex $2$ to vertex $2$?
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![[directed-graph-example.png]]
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Back: $\langle 2, 4, 1, 2, 2 \rangle$ and $\langle 2, 5, 4, 1, 2 \rangle$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723992099163-->
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END%%
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%%ANKI
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Basic
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What are the trails from vertex $2$ to vertex $1$?
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![[undirected-graph-example.png]]
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Back: $\langle 2, 1 \rangle$ and $\langle 2, 5, 1 \rangle$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723992099175-->
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END%%
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### Paths
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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**.
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In computer science, a cycle is sometimes required to have more than one edge:
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* 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.
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* 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.
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%%ANKI
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Basic
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What is a path of (say) graph $G$?
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Back: A trail of $G$ in which no vertex is repeated (except possibly the first and last).
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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).
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<!--ID: 1723992099142-->
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END%%
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%%ANKI
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Basic
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|
What is a cycle of (say) graph $G$?
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Back: A path of $G$ that starts and ends at the same vertex.
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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).
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<!--ID: 1723992829997-->
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END%%
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%%ANKI
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Basic
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What is a trivial cycle of (say) graph $G$?
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Back: A cycle of length $0$.
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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).
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<!--ID: 1723992830003-->
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END%%
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%%ANKI
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Basic
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Which of trails or paths are more general?
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Back: Trails.
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|
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).
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<!--ID: 1723992099152-->
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END%%
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%%ANKI
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Basic
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|
Which of cycles or paths are more general?
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Back: Paths.
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|
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).
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<!--ID: 1723992830005-->
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END%%
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%%ANKI
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Basic
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|
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|
|
Which of cycles or trails are more general?
|
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|
Back: Trails.
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|
|
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).
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<!--ID: 1723992830008-->
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END%%
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%%ANKI
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|
Basic
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What are the paths from vertex $3$ to vertex $6$?
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@ -761,11 +995,11 @@ END%%
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%%ANKI
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Basic
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What are the paths of length $2$ from vertex $2$ to vertex $2$?
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|
What are the paths of length $4$ from vertex $2$ to vertex $2$?
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|
![[directed-graph-example.png]]
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Back: $\langle 2, 2, 2 \rangle$
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Back: $\langle 2, 5, 4, 1, 2 \rangle$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788348-->
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<!--ID: 1723992099168-->
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END%%
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%%ANKI
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@ -779,7 +1013,7 @@ END%%
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%%ANKI
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Basic
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|
What are the paths from vertex $3$ to vertex $6$?
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|
What are the walks from vertex $3$ to vertex $6$?
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|
![[undirected-graph-example.png]]
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Back: $\langle 3, 6 \rangle$, $\langle 3, 6, 3, 6 \rangle$, $\ldots$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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@ -788,71 +1022,16 @@ END%%
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%%ANKI
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Basic
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|
Reachability is a binary relation on what two kinds of objects?
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|
Back: Vertices.
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|
What are the paths from vertex $3$ to vertex $6$?
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|
![[undirected-graph-example.png]]
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Back: $\langle 3, 6 \rangle$
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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|
<!--ID: 1710807788359-->
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<!--ID: 1723992830011-->
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END%%
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%%ANKI
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Basic
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|
How does reachability relate to adjacency?
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Back: Reachability is the transitive generalization of adjacency.
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788364-->
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END%%
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%%ANKI
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Basic
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|
What proximity-based term describes distinct vertices being maximally close?
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|
Back: Adjacency.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788370-->
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END%%
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%%ANKI
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|
Cloze
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{Reachability} is the generalization of {adjacency}.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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|
<!--ID: 1710807788375-->
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END%%
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%%ANKI
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Basic
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|
What does it mean for vertex $u$ to be reachable to vertex $v$?
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|
Back: There exists a path from $u$ to $v$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788379-->
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END%%
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%%ANKI
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Basic
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|
What path must exist in a digraph where vertex $u$ is adjacent to vertex $v$?
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|
Back: $\langle v, u \rangle$
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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|
<!--ID: 1710807788383-->
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END%%
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%%ANKI
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|
Cloze
|
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|
|
Reachable is to paths of length {1:$\geq 0$} whereas adjacency is to paths of length {1:$1$}.
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
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|
|
<!--ID: 1710807788388-->
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|
END%%
|
|
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|
|
|
|
|
|
|
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**.
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%%ANKI
|
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|
Basic
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|
|
What does it mean for a path to be simple?
|
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|
|
Back: All the vertices in the path are distinct.
|
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
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|
<!--ID: 1710807788392-->
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|
END%%
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|
%%ANKI
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|
Basic
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|
|
In a directed graph, when is $\langle v_0, v_1, \ldots, v_k \rangle$ considered a cycle?
|
|
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|
|
In a directed graph, when is path $\langle v_0, v_1, \ldots, v_k \rangle$ considered a non-trivial cycle?
|
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|
|
Back: When $v_0 = v_k$ and there is at least one edge in the path.
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
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|
<!--ID: 1710807788396-->
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|
@ -876,15 +1055,7 @@ END%%
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|
%%ANKI
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|
Basic
|
|
|
|
|
What does it mean for a cycle to be simple?
|
|
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|
|
Back: Except for the first which equals the last, all the vertices in the path are distinct.
|
|
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
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|
<!--ID: 1710807788414-->
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|
END%%
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|
%%ANKI
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|
Basic
|
|
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|
|
How many edges exist in a cycle of a directed graph?
|
|
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|
|
How many edges exist in a non-trivial cycle of a directed graph?
|
|
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|
|
Back: At least one.
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|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
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|
<!--ID: 1710807788421-->
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|
@ -892,7 +1063,7 @@ END%%
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|
%%ANKI
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|
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|
|
Basic
|
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|
|
In an undirected graph, when is $\langle v_0, v_1, \ldots, v_k \rangle$ considered a cycle?
|
|
|
|
|
In an undirected graph, when is $\langle v_0, v_1, \ldots, v_k \rangle$ considered a non-trivial cycle?
|
|
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|
|
Back: When $v_0 = v_k$, $k > 0$, and all edges in the path are distinct.
|
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|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
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|
<!--ID: 1710807788428-->
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|
@ -900,36 +1071,12 @@ END%%
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%%ANKI
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|
Basic
|
|
|
|
|
How many edges exist in a cycle of an undirected graph?
|
|
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|
|
How many edges exist in a non-trivial cycle of an undirected graph?
|
|
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|
|
Back: At least three.
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|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
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|
|
<!--ID: 1710807788435-->
|
|
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|
END%%
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|
%%ANKI
|
|
|
|
|
Cloze
|
|
|
|
|
Path $\langle 1, 2, 4, 1 \rangle$ is not a simple {1:path} but is a simple {1:cycle}.
|
|
|
|
|
![[directed-graph-example.png]]
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|
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|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
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|
|
<!--ID: 1710807788442-->
|
|
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|
END%%
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|
%%ANKI
|
|
|
|
|
Cloze
|
|
|
|
|
Path $\langle 1, 2, 4 \rangle$ is a simple {1:path} but not a simple {1:cycle}.
|
|
|
|
|
![[directed-graph-example.png]]
|
|
|
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
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|
|
<!--ID: 1710807788451-->
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|
END%%
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|
%%ANKI
|
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|
|
Basic
|
|
|
|
|
With respect to paths, what ambiguity exists with the term "simple"?
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|
|
Back: Whether we are referring to simple paths or simple cycles.
|
|
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|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
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|
|
<!--ID: 1710807788458-->
|
|
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|
END%%
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|
%%ANKI
|
|
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|
Basic
|
|
|
|
|
What are the paths to vertex $3$?
|
|
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|
|
@ -950,9 +1097,18 @@ END%%
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|
%%ANKI
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|
Basic
|
|
|
|
|
What are the simple paths of length $1$ to vertex $2$?
|
|
|
|
|
What are the paths of length $1$ to vertex $2$?
|
|
|
|
|
![[directed-graph-example.png]]
|
|
|
|
|
Back: $\langle 1, 2 \rangle$
|
|
|
|
|
Back: $\langle 1, 2 \rangle$ and $\langle 2, 2 \rangle$.
|
|
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|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
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|
|
<!--ID: 1710807788479-->
|
|
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|
END%%
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|
%%ANKI
|
|
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|
|
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).
|
|
|
|
|
<!--ID: 1710807788479-->
|
|
|
|
|
END%%
|
|
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|
|
@ -970,7 +1126,7 @@ END%%
|
|
|
|
|
Basic
|
|
|
|
|
What are the paths of length $2$ to vertex $2$?
|
|
|
|
|
![[directed-graph-example.png]]
|
|
|
|
|
Back: $\langle 4, 1, 2 \rangle$ and $\langle 2, 2, 2 \rangle$
|
|
|
|
|
Back: $\langle 4, 1, 2 \rangle$
|
|
|
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
|
|
|
<!--ID: 1710807788487-->
|
|
|
|
|
END%%
|
|
|
|
|
@ -979,32 +1135,14 @@ END%%
|
|
|
|
|
Basic
|
|
|
|
|
What are the cycles of length $3$ to vertex $2$?
|
|
|
|
|
![[directed-graph-example.png]]
|
|
|
|
|
Back: $\langle 2, 4, 1, 2 \rangle$ and $\langle 2, 2, 2, 2 \rangle$
|
|
|
|
|
Back: $\langle 2, 4, 1, 2 \rangle$
|
|
|
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
|
|
|
<!--ID: 1710807788490-->
|
|
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|
|
END%%
|
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|
%%ANKI
|
|
|
|
|
Basic
|
|
|
|
|
What are the simple cycles of length $3$ to vertex $2$?
|
|
|
|
|
![[directed-graph-example.png]]
|
|
|
|
|
Back: $\langle 2, 4, 1, 2 \rangle$
|
|
|
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
|
|
|
<!--ID: 1710807788493-->
|
|
|
|
|
END%%
|
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|
%%ANKI
|
|
|
|
|
Basic
|
|
|
|
|
What are all the simple cycles containing vertex $2$?
|
|
|
|
|
![[directed-graph-example.png]]
|
|
|
|
|
Back: $\langle 2, 2 \rangle$, $\langle 2, 4, 1, 2 \rangle$, and $\langle 2, 5, 4, 1, 2 \rangle$
|
|
|
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
|
|
|
<!--ID: 1710807788497-->
|
|
|
|
|
END%%
|
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|
%%ANKI
|
|
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|
Basic
|
|
|
|
|
Why isn't $\langle 3, 6, 3 \rangle$ considered a cycle?
|
|
|
|
|
*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).
|
|
|
|
|
@ -1022,7 +1160,7 @@ END%%
|
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|
|
%%ANKI
|
|
|
|
|
Basic
|
|
|
|
|
What are the simple paths to vertex $2$?
|
|
|
|
|
What are the paths to vertex $2$?
|
|
|
|
|
![[undirected-graph-example.png]]
|
|
|
|
|
Back: $\langle 2 \rangle$, $\langle 1, 2 \rangle$, $\langle 5, 2 \rangle$, $\langle 1, 5, 2 \rangle$, $\langle 5, 1, 2 \rangle$
|
|
|
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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@ -1031,18 +1169,18 @@ END%%
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%%ANKI
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Basic
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What are the simple cycles containing vertex $2$?
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What are the cycles to vertex $2$?
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![[undirected-graph-example.png]]
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Back: $\langle 2, 5, 1, 2 \rangle$ and $\langle 2, 1, 5, 2 \rangle$
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Back: $\langle 2 \rangle$, $\langle 2, 5, 1, 2 \rangle$ and $\langle 2, 1, 5, 2 \rangle$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788519-->
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END%%
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%%ANKI
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Basic
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What are the cycles containing vertex $3$?
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What are the cycles to vertex $3$?
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![[undirected-graph-example.png]]
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Back: N/A
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Back: $\langle 3 \rangle$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788525-->
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END%%
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@ -1050,7 +1188,7 @@ END%%
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%%ANKI
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Basic
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What does it mean for a graph to be acyclic?
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Back: It has no simple cycles.
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Back: It has no cycles.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1710807788532-->
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END%%
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