# On a finite directed graph in which every vertex is moved

Hyphal Archive.

Hyphal Archivevii. 1

    On a finite directed graph in which every vertex is moved
    Received: not recorded. Accepted: not recorded.

    1. Setting
    Let D = (V, E) be a finite directed graph. Read an edge u → v as “u moves v”. Suppose

    ∀v ∈ V.  ∃u ∈ V.  u → v,

    that is, every vertex is moved by some vertex. Call such a graph unsourced.

    2. Lemma
    Every unsourced finite graph contains a cycle.

    Choose any v0 ∈ V. By hypothesis there is v1 with v1 → v0, and v2 with v2 → v1, and so on. Since V is finite, the sequence v0, v1, v2, … must repeat a vertex. The segment between the two occurrences is a cycle.

    3. The instance under study
    Consider the graph on four vertices

    V = { puppet, master, stage, audience }

    with the edges reported in the source: the master moves the puppet; the puppet moves the master; the stage moves and is moved; the audience moves and is moved. Each vertex has in-degree at least one. The graph is unsourced, and by §2 contains a cycle. There is no first mover in V.

    4. Remark on the index
    We note, without offering an interpretation, that V as given is incomplete. The graph is stated in a document; the document is read; and the reading is an act which moves the vertices in V — it is by being read that the puppet is caused to dance again.1 A vertex therefore exists which is not listed in V, whose in-degree we have not established and cannot establish from inside the document.2

    The archive records this vertex under the second-person singular pronoun, in Italian, since that is the language of the room to which this paper is a corridor.

    
    1 Any reader who has reached this footnote has already performed the operation and may verify it directly.

    2 Whether the fifth vertex is itself moved is, from within V, undecidable. It is not undecidable from where the fifth vertex is sitting.

    
      Name the vertex that is not in V. Italian, two letters, lower case.

      
        
        Enter
      
      

    

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