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Type | Label | Description |
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Statement | ||
Theorem | umgr1wlknloop 39801* | In a multigraph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 3-Jan-2021.) |
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Theorem | wlkRes 39802* | Restrictions of walks (i.e. special kinds of walks, for example paths - see pthsfval 39851) are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 30-Dec-2020.) (Proof shortened by AV, 15-Jan-2021.) |
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Theorem | wlkson 39803* | The set of walks between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 30-Dec-2020.) (Proof shortened by AV, 15-Jan-2021.) |
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Theorem | iswlkOn 39804 | Properties of a pair of functions to be a walk between two given vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Revised by AV, 31-Dec-2020.) (Proof shortened by AV, 31-Jan-2021.) |
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Theorem | wlkOnprop 39805 | Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 31-Dec-2020.) (Proof shortened by AV, 16-Jan-2021.) |
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Theorem | 1wlkepvtx 39806 | The endpoints of a walk are vertices. (Contributed by AV, 31-Jan-2021.) |
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Theorem | wlkOniswlk 39807 | A walk between two vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 2-Jan-2021.) |
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Theorem | wlkOnwlk 39808 | A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.) |
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Theorem | wlkOnwlk1l 39809 | A walk is a walk from its first vertex to its last vertex. (Contributed by AV, 7-Feb-2021.) |
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Theorem | 2Wlklem 39810* | Lemma for upgr2wlk 39811 and 2wlklemA 25340. Identical with is2wlk 25351. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
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Theorem | upgr2wlk 39811 | Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) |
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Theorem | red1wlklem 39812 | Lemma for red1wlk 39813. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.) |
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Theorem | red1wlk 39813 | A 1-walk ending at the last but one vertex of the walk is a 1-walk. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.) |
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For a hypergraph, the property to be "loop-free" is expressed by
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Theorem | umgrislfupgrlem 39814 | Lemma for umgrislfupgr 39815 and usgrislfuspgr 39816. (Contributed by AV, 27-Jan-2021.) |
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Theorem | umgrislfupgr 39815* | A multigraph is a loop-free pseudograph. (Contributed by AV, 27-Jan-2021.) |
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Theorem | usgrislfuspgr 39816* | A simple graph is a loop-free simple pseudograph. (Contributed by AV, 27-Jan-2021.) |
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Theorem | 1wlkvtxeledg 39817* | Each pair of adjacent vertices in a 1-walk is a subset of an edge. (Contributed by AV, 28-Jan-2021.) |
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Theorem | lfgrwlkprop 39818* | Two adjacent vertices in a 1-walk are different in a loop-free graph. (Contributed by AV, 28-Jan-2021.) |
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Theorem | lfgriswlk 39819* | Conditions for a pair of functions to be a 1-walk in a loop-free graph. (Contributed by AV, 28-Jan-2021.) |
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Theorem | lfgr1wlknloop 39820* | In a loop-free graph, each walk has no loops! (Contributed by AV, 2-Feb-2021.) |
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Theorem | 1wlkdlem1 39821* | Lemma 1 for 1wlkd 39825. (Contributed by AV, 7-Feb-2021.) |
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Theorem | 1wlkdlem2 39822* | Lemma 2 for 1wlkd 39825. (Contributed by AV, 7-Feb-2021.) |
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Theorem | 1wlkdlem3 39823* | Lemma 3 for 1wlkd 39825. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
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Theorem | 1wlkdlem4 39824* | Lemma 4 for 1wlkd 39825. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 23-Jan-2021.) |
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Theorem | 1wlkd 39825* | Two words representing a walk in a graph. (Contributed by AV, 7-Feb-2021.) |
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Syntax | ctrls 39826 | Extend class notation with trails (within a graph). |
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Syntax | ctrlson 39827 | Extend class notation with tails between two vertices (within a graph). |
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Definition | df-trls 39828* |
Define the set of all Trails (in an undirected graph).
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct. According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5. Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
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Definition | df-trlson 39829* | Define the collection of trails with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
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Theorem | trlsfval 39830* | The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Proof shortened by AV, 31-Jan-2021.) |
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Theorem | isTrl 39831 | Conditions for a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Proof shortened by AV, 31-Jan-2021.) |
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Theorem | upgrtrls 39832* | The set of trails in a pseudograph, definition of walks expanded. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) |
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Theorem | upgristrl 39833* | Properties of a pair of functions to be a trail in a pseudograph, definition of walks expanded. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) |
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Theorem | upgrf1istrl 39834* | Properties of a pair of a one-to-one function into the set of indices of edges and a function into the set of vertices to be a trail in a pseudograph. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) |
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Theorem | trlis1wlk 39835 | A trail is a walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) |
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Theorem | 1wlksonproplem 39836* | Lemma for theorems for properties of walks between two vertices, e.g. trlsonprop 39839. (Contributed by AV, 16-Jan-2021.) |
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Theorem | trlsonfval 39837* | The set of trails between two vertices. (Contributed by Alexander van der Vekens, 4-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 15-Jan-2021.) |
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Theorem | istrlson 39838 | Properties of a pair of functions to be a trail between two given vertices. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Revised by AV, 7-Jan-2021.) |
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Theorem | trlsonprop 39839 | Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 16-Jan-2021.) |
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Theorem | trlsonistrl 39840 | A trail between two vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 7-Jan-2021.) |
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Theorem | trlsonwlkon 39841 | A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.) |
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Theorem | trlOntrl 39842 | A trail is a trail between its endpoints. (Contributed by AV, 31-Jan-2021.) |
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Syntax | cpths 39843 | Extend class notation with paths (of a graph). |
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Syntax | cspths 39844 | Extend class notation with simple paths (of a graph). |
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Syntax | cpthson 39845 | Extend class notation with paths between two vertices (within a graph). |
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Syntax | cspthson 39846 | Extend class notation with simple paths between two vertices (within a graph). |
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Definition | df-pths 39847* |
Define the set of all paths (in an undirected graph).
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices." According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see wlkdvspth 25394). Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
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Definition | df-spths 39848* |
Define the set of all simple paths (in an undirected graph).
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices." Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
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Definition | df-pthson 39849* | Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
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Definition | df-spthson 39850* | Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 9-Jan-2021.) |
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Theorem | pthsfval 39851* | The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.) |
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Theorem | spthsfval 39852* | The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.) |
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Theorem | isPth 39853 | Conditions for a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.) |
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Theorem | issPth 39854 | Conditions for a pair of functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.) |
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Theorem | PthisTrl 39855 | A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
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Theorem | sPthisPth 39856 | A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
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Theorem | pthis1wlk 39857 | A path is a 1-walk (in an undirected graph). (Contributed by AV, 6-Feb-2021.) |
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Theorem | pthdivtx 39858 | The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.) |
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Theorem | pthdadjvtx 39859 | The adjacent vertices of a path of length at least 2 are distinct. (Contributed by AV, 5-Feb-2021.) |
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Theorem | 2pthnloop 39860* | A path of length at least 2 does not contain a loop. In contrast, a path of length 1 can contain/be a loop, see lppthon 39962. (Contributed by AV, 6-Feb-2021.) |
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Theorem | upgr2pthnlp 39861* | A path of length at least 2 in a pseudograph does not contain a loop. (Contributed by AV, 6-Feb-2021.) |
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Theorem | spthdep 39862 | A simple path (at least of length 1) has different start and end points (in an undirected graph). (Contributed by AV, 31-Jan-2021.) |
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Theorem | pthdepissPth 39863 | A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 12-Jan-2021.) |
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Theorem | upgrwlkdvdelem 39864* | Lemma for upgrwlkdvde 39865. Formerly wlkdvspthlem 25393. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Proof shortened by AV, 17-Jan-2021.) |
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Theorem | upgrwlkdvde 39865 | In a pseudograph, all edges of a walk consisting of different vertices are different. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspths1wlk 39866. (Contributed by AV, 17-Jan-2021.) |
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Theorem | upgrspths1wlk 39866* | The set of simple paths in a pseudograph, expressed as walk. Notice that this theorem would not hold for arbitrary hypergraphs, since a walk with distinct vertices does not need to be a trail: let E = { p0, p1, p2 } be a hyperedge, then ( p0, e, p1, e, p2 ) is walk with distinct vertices, but not with distinct edges. Therefore, E is not a trail and, by definition, also no path. (Contributed by AV, 11-Jan-2021.) (Proof shortened by AV, 17-Jan-2021.) |
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Theorem | upgrwlkdvspth 39867 | A walk consisting of different vertices is a simple path. Formerly wlkdvspth 25394. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspths1wlk 39866. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Revised by AV, 17-Jan-2021.) |
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Theorem | pthsonfval 39868* | The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) |
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Theorem | spthson 39869* | The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) |
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Theorem | ispthson 39870 | Properties of a pair of functions to be a path between two given vertices. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) |
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Theorem | isspthson 39871 | Properties of a pair of functions to be a simple path between two given vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) |
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Theorem | pthsonprop 39872 | Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 16-Jan-2021.) |
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Theorem | spthonprop 39873 | Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) |
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Theorem | pthonispth-av 39874 | A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 17-Jan-2021.) |
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Theorem | pthontrlon 39875 | A path between two vertices is a trail between these vertices. (Contributed by AV, 24-Jan-2021.) |
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Theorem | pthOnpth 39876 | A path is a path between its endpoints. (Contributed by AV, 31-Jan-2021.) |
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Theorem | isspthonpth-av 39877 | A pair of functions is a simple path between two given vertices iff it is a simple path starting and ending at the two vertices. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-Jan-2021.) |
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Theorem | spthonisspth-av 39878 | A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 18-Jan-2021.) |
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Theorem | spthonpthon 39879 | A simple path between two vertices is a path between these vertices. (Contributed by AV, 24-Jan-2021.) |
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Theorem | spthonepeq-av 39880 | The endpoints of a simple path between two vertices are equal iff the path is of length 0. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 18-Jan-2021.) |
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Theorem | uhgr1wlkspthlem1 39881 | Lemma 1 for uhgr1wlkspth 39883. (Contributed by AV, 25-Jan-2021.) |
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Theorem | uhgr1wlkspthlem2 39882 | Lemma 2 for uhgr1wlkspth 39883. (Contributed by AV, 25-Jan-2021.) |
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Theorem | uhgr1wlkspth 39883 | Any walk of length 1 between two different vertices is a simple path. (Contributed by AV, 25-Jan-2021.) |
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Theorem | usgr2wlkneq 39884 | The vertices and edges are pairwise different in a walk of length 2 in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.) |
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Theorem | usgr2wlkspthlem1 39885 | Lemma 1 for usgr2wlkspth 39887. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.) |
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Theorem | usgr2wlkspthlem2 39886 | Lemma 2 for usgr2wlkspth 39887. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.) |
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Theorem | usgr2wlkspth 39887 | In a simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.) |
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Theorem | pthdlem1 39888* | Lemma 1 for pthd 39891. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 9-Feb-2021.) |
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Theorem | pthdlem2lem 39889* | Lemma for pthdlem2 39890. (Contributed by AV, 10-Feb-2021.) |
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Theorem | pthdlem2 39890* | Lemma 2 for pthd 39891. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 10-Feb-2021.) |
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Theorem | pthd 39891* | Two words representing a trail which also represent a path in a graph. (Contributed by AV, 10-Feb-2021.) |
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Syntax | cclwlks 39892 | Extend class notation with closed walks (of a graph). |
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Definition | df-clwlks 39893* |
Define the set of all closed walks (in an undirected graph).
According to definition 4 in [Huneke] p. 2: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0". According to the definition of a walk as two mappings f from { 0 , ... , ( n - 1 ) } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed walk is represented by the following sequence: p(0) e(f(0)) p(1) e(f(1)) ... p(n-1) e(f(n-1)) p(n)=p(0). Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 25549! (Contributed by Alexander van der Vekens, 12-Mar-2018.) (Revised by AV, 16-Feb-2021.) |
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Theorem | clwlkS 39894* | The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) |
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Theorem | isclWlk 39895 | Properties of a pair of functions to be a closed walk (in an undirected graph) in terms of walks. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) |
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Theorem | isclWlkb 39896 | Generalisation of isclwlk 25540: A pair of functions represents a closed walk iff it represents a walk in which the first vertex is equal to the last vertex. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.) |
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Theorem | clwlkis1wlk 39897 | A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) |
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Theorem | clwlk1wlk 39898 | Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 16-Feb-2021.) |
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Theorem | clwlks1wlks 39899 | Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Feb-2021.) |
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Theorem | isclWlke 39900* | Properties of a pair of functions to be a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.) |
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