From c31e0c11c66a6d18bdc128959899da83391206e1 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Sun, 18 Aug 2024 09:04:52 -0600 Subject: [PATCH] Distinguish walks, trails, and paths. --- .../plugins/obsidian-to-anki-plugin/data.json | 20 +- notes/_journal/2024-08-18.md | 11 + notes/hashing/addressing.md | 2 +- notes/ontology/philosophy/index.md | 87 ---- notes/ontology/philosophy/properties.md | 99 ++++ notes/set/graphs.md | 426 ++++++++++++------ 6 files changed, 408 insertions(+), 237 deletions(-) create mode 100644 notes/_journal/2024-08-18.md create mode 100644 notes/ontology/philosophy/properties.md diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 035629a..c4bcdb6 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -178,7 +178,15 @@ "function-injective.png", "function-surjective.png", "function-general.png", - "function-kernel.png" + "function-kernel.png", + "closed-addressing.png", + "open-addressing.png", + "directed-graph-example.png", + "undirected-graph-example.png", + "graph-isomorphic.png", + "graph-induced-subgraph.png", + "graph-subgraph.png", + "graph-non-subgraph.png" ], "File Hashes": { "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8", @@ -359,7 +367,7 @@ "_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b", "set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53", "set/index.md": "c103501e345a1b8201a26f2e83ed8379", - "set/graphs.md": "f0cd201673f2a999321dda6c726e8734", + "set/graphs.md": "ce7030d3bae17b56319e8b5526a63c95", "_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb", "_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391", "awk/variables.md": "e40a20545358228319f789243d8b9f77", @@ -632,7 +640,7 @@ "_journal/2024-07-13.md": "13b5101306b5542b8a1381a6477378ca", "_journal/2024-07/2024-07-12.md": "6603ed8a3f9a9e87bf40e81b03e96356", "hashing/static.md": "3ec6eaee73fb9b599700f5a56b300b83", - "hashing/addressing.md": "583da2fc0320a2f1daacc0e01ca66d83", + "hashing/addressing.md": "01b33abe25aae285e1641fa43470065b", "ontology/index.md": "0994403dcd84415f1459752129b55f65", "ontology/permissivism.md": "643e815a79bc5c050cde9f996aa44ef5", "ontology/properties.md": "91ece501551c444afcd119d7197958ef", @@ -687,7 +695,7 @@ "ontology/rdf/index.md": "36424c9bad6088cdee67f74e3b8a019f", "ontology/philosophy/permissivism.md": "643e815a79bc5c050cde9f996aa44ef5", "ontology/philosophy/nominalism.md": "46245c644238157e15c7cb6def27d90a", - "ontology/philosophy/index.md": "6c7c60f91f78fdc1cdd8c012b1ac4ebd", + "ontology/philosophy/index.md": "d132b8f4a69bdb664c822366fb27fa64", "ontology/philosophy/dialetheism.md": "56dd05b38519f90c5cab93637978b3b3", "_journal/2024-07-29.md": "a480e577b06a94755b6ebf4ac9ee5732", "_journal/2024-07/2024-07-28.md": "ff5dcfb3dc1b5592894363414e20b02f", @@ -730,7 +738,9 @@ "_journal/2024-08/2024-08-15.md": "7c3a96a25643b62b0064bf32cb17d92f", "_journal/2024-08-17.md": "b06a551560c377f61a1b39286cd43cee", "_journal/2024-08/2024-08-16.md": "096d9147a9e3e7a947558f8dec763a2c", - "set/order.md": "373f4336d4845a3c2188d2215ac5fbc4" + "set/order.md": "373f4336d4845a3c2188d2215ac5fbc4", + "_journal/2024-08-18.md": "240ce6377b91d977f4fedc30724891f6", + "ontology/philosophy/properties.md": "41b32249d3e4c23d73ddb3a417d65a4c" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-08-18.md b/notes/_journal/2024-08-18.md new file mode 100644 index 0000000..2751023 --- /dev/null +++ b/notes/_journal/2024-08-18.md @@ -0,0 +1,11 @@ +--- +title: "2024-08-18" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Distinguish between walks, trails, and paths. \ No newline at end of file diff --git a/notes/hashing/addressing.md b/notes/hashing/addressing.md index ff099d3..992ea29 100644 --- a/notes/hashing/addressing.md +++ b/notes/hashing/addressing.md @@ -596,7 +596,7 @@ END%% %%ANKI Cloze -{Probing} refers to the {sequential examining of slots} performed in open addressing. +{Probing} refers to the {iterative examining of slots} performed in open addressing. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% diff --git a/notes/ontology/philosophy/index.md b/notes/ontology/philosophy/index.md index de354ca..b3d613d 100644 --- a/notes/ontology/philosophy/index.md +++ b/notes/ontology/philosophy/index.md @@ -117,93 +117,6 @@ Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Pres END%% -## Properties - -A **property** is an entity that can be predicated of things or, in other words, attributed to them. - -%%ANKI -Basic -What is a property? -Back: An entity that can be predicated or attributed to things. -Reference: Francesco Orilia and Michele Paolini Paoletti, “Properties,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta, Spring 2022 (Metaphysics Research Lab, Stanford University, 2022), [https://plato.stanford.edu/archives/spr2022/entries/properties/](https://plato.stanford.edu/archives/spr2022/entries/properties/). - -END%% - -### Instantiation - -An entity is said to **instantiate** a property if said entity bears a connection to the property. For example, a human instantiates the property of *being human* and a man instantiates the properties of *being human* and *being a man*. - -%%ANKI -Basic -What is instantiation? -Back: A relation held between an entity and the properties that describe the entity. -Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). - -END%% - -%%ANKI -Cloze -A man is said to {instantiate} the property of *being a man*. -Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). - -END%% - -%%ANKI -Basic -What is self-instantiation? -Back: The instantiation of a property by itself. -Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). - -END%% - -%%ANKI -Basic -What is non-self-instantiation? -Back: The non-instantiation of a property by itself. -Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). - -END%% - -%%ANKI -Basic -Suppose all properties are self-instantiating. What must be said about *being honest*? -Back: The property *being honest* is honest. -Reference: Francesco Orilia and Michele Paolini Paoletti, “Properties,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta, Spring 2022 (Metaphysics Research Lab, Stanford University, 2022), [https://plato.stanford.edu/archives/spr2022/entries/properties/](https://plato.stanford.edu/archives/spr2022/entries/properties/). - -END%% - -%%ANKI -Basic -Suppose properties are abstracta. What self-instantiation is thus formed? -Back: The property of abstractness is abstract. -Reference: Francesco Orilia and Michele Paolini Paoletti, “Properties,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta, Spring 2022 (Metaphysics Research Lab, Stanford University, 2022), [https://plato.stanford.edu/archives/spr2022/entries/properties/](https://plato.stanford.edu/archives/spr2022/entries/properties/). - -END%% - -%%ANKI -Basic -What is the paradox of non-self-instantiation? -Back: The property *non-self-instantiation* is non-self-instantiating iff it is self-instantiating. -Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). - -END%% - -%%ANKI -Basic -Let $P$ be the property *is non-self-instantiating*. What happens if $P$ is non-self-instantiating? -Back: Then $P$ must be self-instantiating. -Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). - -END%% - -%%ANKI -Basic -Let $P$ be the property *is non-self-instantiating*. What happens if $P$ is self-instantiating? -Back: Then $P$ must be non-self-instantiating. -Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). - -END%% - ## Bibliography * Francesco Orilia and Michele Paolini Paoletti, “Properties,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta, Spring 2022 (Metaphysics Research Lab, Stanford University, 2022), [https://plato.stanford.edu/archives/spr2022/entries/properties/](https://plato.stanford.edu/archives/spr2022/entries/properties/). diff --git a/notes/ontology/philosophy/properties.md b/notes/ontology/philosophy/properties.md new file mode 100644 index 0000000..1de65bd --- /dev/null +++ b/notes/ontology/philosophy/properties.md @@ -0,0 +1,99 @@ +--- +title: Properties +TARGET DECK: Obsidian::H&SS +FILE TAGS: ontology::philosophy +tags: + - ontology +--- + +## Overview + +A **property** is an entity that can be predicated of things or, in other words, attributed to them. + +%%ANKI +Basic +What is a property? +Back: An entity that can be predicated or attributed to things. +Reference: Francesco Orilia and Michele Paolini Paoletti, “Properties,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta, Spring 2022 (Metaphysics Research Lab, Stanford University, 2022), [https://plato.stanford.edu/archives/spr2022/entries/properties/](https://plato.stanford.edu/archives/spr2022/entries/properties/). + +END%% + +### Instantiation + +An entity is said to **instantiate** a property if said entity bears a connection to the property. For example, a human instantiates the property of *being human* and a man instantiates the properties of *being human* and *being a man*. + +%%ANKI +Basic +What is instantiation? +Back: A relation held between an entity and the properties that describe the entity. +Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). + +END%% + +%%ANKI +Cloze +A man is said to {instantiate} the property of *being a man*. +Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). + +END%% + +%%ANKI +Basic +What is self-instantiation? +Back: The instantiation of a property by itself. +Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). + +END%% + +%%ANKI +Basic +What is non-self-instantiation? +Back: The non-instantiation of a property by itself. +Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). + +END%% + +%%ANKI +Basic +Suppose all properties are self-instantiating. What must be said about *being honest*? +Back: The property *being honest* is honest. +Reference: Francesco Orilia and Michele Paolini Paoletti, “Properties,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta, Spring 2022 (Metaphysics Research Lab, Stanford University, 2022), [https://plato.stanford.edu/archives/spr2022/entries/properties/](https://plato.stanford.edu/archives/spr2022/entries/properties/). + +END%% + +%%ANKI +Basic +Suppose properties are abstracta. What self-instantiation is thus formed? +Back: The property of abstractness is abstract. +Reference: Francesco Orilia and Michele Paolini Paoletti, “Properties,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta, Spring 2022 (Metaphysics Research Lab, Stanford University, 2022), [https://plato.stanford.edu/archives/spr2022/entries/properties/](https://plato.stanford.edu/archives/spr2022/entries/properties/). + +END%% + +%%ANKI +Basic +What is the paradox of non-self-instantiation? +Back: The property *non-self-instantiation* is non-self-instantiating iff it is self-instantiating. +Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). + +END%% + +%%ANKI +Basic +Let $P$ be the property *is non-self-instantiating*. What happens if $P$ is non-self-instantiating? +Back: Then $P$ must be self-instantiating. +Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). + +END%% + +%%ANKI +Basic +Let $P$ be the property *is non-self-instantiating*. What happens if $P$ is self-instantiating? +Back: Then $P$ must be non-self-instantiating. +Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). + +END%% + +## Bibliography + +* Francesco Orilia and Michele Paolini Paoletti, “Properties,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta, Spring 2022 (Metaphysics Research Lab, Stanford University, 2022), [https://plato.stanford.edu/archives/spr2022/entries/properties/](https://plato.stanford.edu/archives/spr2022/entries/properties/). +* Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). \ No newline at end of file diff --git a/notes/set/graphs.md b/notes/set/graphs.md index f34ae99..2b2f823 100644 --- a/notes/set/graphs.md +++ b/notes/set/graphs.md @@ -655,21 +655,88 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition ( END%% -## Paths +### Handshake Lemma -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. +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 -Let $G = \langle V, E \rangle$ be a graph. What *is* a path from vertex $u$ to vertex $v$? -Back: A sequence of vertices $\langle u, \ldots, v \rangle$ such that there is an edge for each consecutive pair of vertices. +*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 path $\langle v_0, v_1, \ldots, v_k \rangle$. What is the path's length? +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). @@ -677,23 +744,23 @@ END%% %%ANKI Basic -In terms of edges, what is the length of a path? -Back: The number of edges specified in the path. +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 path? -Back: One less than the number of vertices specified in the path. +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 path of $G$ is said to contain what? +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). @@ -701,28 +768,195 @@ END%% %%ANKI Basic -How does a path of a graph relate to the concept of adjacency? -Back: Each vertex must be adjacent to the vertex preceding it in the path. +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 path of a directed graph relate to the concept of incidence? -Back: There exists an edge incident to each vertex that is also incident from the vertex preceding it in the path. +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 path of an undirected graph relate to the concept of incidence? -Back: There exists an edge incident on each vertex and the vertex preceding it in the path. +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: $\langle 2, 4, 1, 2, 2 \rangle$ 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 and 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 $0$. +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 $6$? @@ -761,11 +995,11 @@ END%% %%ANKI Basic -What are the paths of length $2$ from vertex $2$ to vertex $2$? +What are the paths of length $4$ from vertex $2$ to vertex $2$? ![[directed-graph-example.png]] -Back: $\langle 2, 2, 2 \rangle$ +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 @@ -779,7 +1013,7 @@ END%% %%ANKI Basic -What are the paths from vertex $3$ to vertex $6$? +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). @@ -788,71 +1022,16 @@ END%% %%ANKI Basic -Reachability is a binary relation on what two kinds of objects? -Back: Vertices. +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 -How does reachability relate to adjacency? -Back: Reachability is the transitive generalization of adjacency. -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Basic -What proximity-based term describes distinct vertices being maximally close? -Back: Adjacency. -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Cloze -{Reachability} is the generalization of {adjacency}. -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Basic -What does it mean for vertex $u$ to be reachable to vertex $v$? -Back: There exists a path from $u$ to $v$. -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Basic -What path must exist in a digraph where vertex $u$ is adjacent to vertex $v$? -Back: $\langle v, u \rangle$ -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Cloze -Reachable is to paths of length {1:$\geq 0$} whereas adjacency is to paths of length {1:$1$}. -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -A path is **simple** if all vertices in the path are distinct. In a directed graph, path $\langle v_0, v_1, \ldots, v_k \rangle$ forms a **cycle** if $v_0 = v_k$ and the path contains at least one edge. In an undirected graph, path $\langle v_0, v_1, \ldots, v_k \rangle$ forms a cycle if $v_0 = v_k$ and all edges are distinct. We say a cycle is **simple** if all vertices in the path (barring the first and last) are distinct. A graph with no simple cycles is **acyclic**. - -%%ANKI -Basic -What does it mean for a path to be simple? -Back: All the vertices in the path are distinct. -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Basic -In a directed graph, when is $\langle v_0, v_1, \ldots, v_k \rangle$ considered a cycle? +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). @@ -876,15 +1055,7 @@ END%% %%ANKI Basic -What does it mean for a cycle to be simple? -Back: Except for the first which equals the last, all the vertices in the path are distinct. -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Basic -How many edges exist in a cycle of a directed graph? +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). @@ -892,7 +1063,7 @@ END%% %%ANKI Basic -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? 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). @@ -900,36 +1071,12 @@ END%% %%ANKI Basic -How many edges exist in a cycle of an undirected graph? +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 -Cloze -Path $\langle 1, 2, 4, 1 \rangle$ is not a simple {1:path} but is a simple {1:cycle}. -![[directed-graph-example.png]] -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Cloze -Path $\langle 1, 2, 4 \rangle$ is a simple {1:path} but not a simple {1:cycle}. -![[directed-graph-example.png]] -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Basic -With respect to paths, what ambiguity exists with the term "simple"? -Back: Whether we are referring to simple paths or simple cycles. -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - %%ANKI Basic What are the paths to vertex $3$? @@ -950,9 +1097,18 @@ END%% %%ANKI 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 $\langle 2, 2 \rangle$. +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%% @@ -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). 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). END%% %%ANKI Basic -What are the simple cycles of length $3$ to vertex $2$? -![[directed-graph-example.png]] -Back: $\langle 2, 4, 1, 2 \rangle$ -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Basic -What are all the simple cycles containing vertex $2$? -![[directed-graph-example.png]] -Back: $\langle 2, 2 \rangle$, $\langle 2, 4, 1, 2 \rangle$, and $\langle 2, 5, 4, 1, 2 \rangle$ -Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). - -END%% - -%%ANKI -Basic -Why isn't $\langle 3, 6, 3 \rangle$ considered a cycle? +*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%% %%ANKI Basic -What are the simple paths to vertex $2$? +What are the paths to vertex $2$? ![[undirected-graph-example.png]] Back: $\langle 2 \rangle$, $\langle 1, 2 \rangle$, $\langle 5, 2 \rangle$, $\langle 1, 5, 2 \rangle$, $\langle 5, 1, 2 \rangle$ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). @@ -1031,18 +1169,18 @@ END%% %%ANKI Basic -What are the simple cycles containing vertex $2$? +What are the cycles to vertex $2$? ![[undirected-graph-example.png]] -Back: $\langle 2, 5, 1, 2 \rangle$ and $\langle 2, 1, 5, 2 \rangle$ +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 containing vertex $3$? +What are the cycles to vertex $3$? ![[undirected-graph-example.png]] -Back: N/A +Back: $\langle 3 \rangle$ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% @@ -1050,7 +1188,7 @@ END%% %%ANKI Basic What does it mean for a graph to be acyclic? -Back: It has no simple cycles. +Back: It has no cycles. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%%