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- At 13:43, pole 18-IV. (Not really a pole but a plate on the ground.)
- At 14:14, pole 21.
- At 14:46, pole 22-I.

FiniteDirectedGraphs = { { "V" : V, "E" : E } | V in FiniteSets and E subset { (v1, v2) | v1 in V and v2 in V } }To label the vertices, we first need to define the set of all functions over a given domain:

FiniteFunctionsForDomain(D in FiniteSets) = { F | size F = size D } where F = { { d : c } | d in D and c in Any}With this we can define the set of all labled directed graphs:

LabledFiniteDirectedGraphs = { { "G" : G, "L" : L } | G in FiniteDirectedGraphs and L in FiniteFunctionsForDomain(G."V") }Now we can define a kind of product of two labled directed graphs, like:

Matching(G1 in LabledFiniteDirectedGraphs, G2 in LabledFiniteDirectedGraphs) = { "G" : { "V" : { (v1,v2) | v1 in G1."G"."V" and v2 in G2."G"."V" and G1."L".v1 = G2."L".v2 }, "E" : { ((v11,v12), (v21,v22)) | (v11,v21) in G1."E" and (v12,v22) in G2."E" } }, "L" : { (v1,v2) : G1."L".v1 } } in LabledFiniteDirectedGraphsI would like to define something as sequence equivalence, where the sequences of lables that can be produced by any walk over a labled directed graph is equal to that of another graph. First we define the set of all directed paths of a directed graph and with this we define the sequences:

DirectedPaths(G in DirectedGraphs) = { (v[1], .., v[n]) | ForEach i in NatNum (1 <= i and i < n implies (v[i],v[i+1]) in G."E") } Sequences(G in LabledFiniteDirectedGraphs) = { (G."L".v[1], .., G."L".v[n]) | (v[1], .., v[n]) in DirectedPaths(G."G") }Now we can define what it means that two graphs a sequence equivalent:

SequenceEquivalent(G1 in LabledFiniteDirectedGraphs, G2 in LabledFiniteDirectedGraphs) = Sequences(G1) = Sequences(G2)Now, one could argue that such a test is impossible to execute, because one would have to compare an infinite set of sequences, but I believe it is sufficient to test this for all sequences that are one longer than the largest number of vertices of one of the two graphs. Maybe, there are even beter ways to check the equivalence, for example, if one could construct a correspondance relationship between the two set of vertices and somehow prove that the graphs are isomorphic with respect to the lables of the vertices. The algorithm, I am looking for, is an algorithm that finds a smallest sequence equivalent labled directed graph of given labled direct graph, where smallest is defined by the number of vertices.

- At 11:53, pole 24-I.
- At 12:02, pole 24.
- At 12:22, pole 26.
- At 12:41, pole 26-III.
- At 12:50, pole 26-II.
- At 13:13, pole 26-I.

CyclicDirectedPaths(G in DirectedGraphs) = { (v[1], .., v[n]) | ForEach i in NatNum (1 <= i and i < n implies (v[i],v[i+1]) in G."E") and (v[n],v[1]) in G."E" } Sequences(G in LabledFiniteDirectedGraphs) = { (G."L".v[1], .., G."L".v[n]) | (v[1], .., v[n]) in CyclicDirectedPaths(G."G") }(I have to admit that I am still wrestling with the syntax and the semantics I am using in these specifications. The idea is that it should be close to some mathematical notations, but these notations are often hard to interpret. The equal sign in the above notation can both mean equality and definition depending on the context. Maybe it is better to use a more explicit syntax for definition using the keyword

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