Description of graph6, sparse6 and digraph6 encodings
-----------------------------------------------------
Brendan McKay, brendan.mckay@anu.edu.au
Updated Jun 2015, Apr 2022, Aug 2023.

General principles:

  All numbers in this description are in decimal unless obviously 
  in binary.

  The character code used in all cases is ASCII, but no bytes are
  present which would cause ambiguity if the file is read as UTF8.

  Apart from the header, there is one object per line. Apart from
  the header, end-of-line characters, and the characters ":", ";"
  and "&" which might start a line, all bytes have a value in the
  range 63-126 (which are all printable ASCII characters). A file of
  objects is a text file, so whatever end-of-line convention is
  locally used is fine; however the C library input routines must
  show the standard single-LF end of line to programs).

Bit vectors:

  A bit vector x of length k can be represented as follows.  
      Example:  1000101100011100

  (1) Pad on the right with 0 to make the length a multiple of 6.
      Example:  100010110001110000

  (2) Split into groups of 6 bits each.
      Example:  100010 110001 110000

  (3) Add 63 to each group, considering them as bigendian binary numbers.
      Example:  97 112 111

  These values are then stored one per byte.  
  So, the number of bytes is ceiling(k/6).

  Let R(x) denote this representation of x as a string of bytes.
      
Small nonnegative integers:
 
  Let n be an integer in the range 0-68719476735 (2^36-1).

  If 0 <= n <= 62, define N(n) to be the single byte n+63.
  If 63 <= n <= 258047, define N(n) to be the four bytes
      126 R(x), where x is the bigendian 18-bit binary form of n.
  If 258048 <= n <= 68719476735, define N(n) to be the eight bytes
      126 126 R(x), where x is the bigendian 36-bit binary form of n.

  Examples:  N(30) = 93
             N(12345) = N(000011 000000 111001) = 126 66 63 120
             N(460175067) = N(000000 011011 011011 011011 011011 011011)
                          = 126 126 63 90 90 90 90 90


Description of graph6 format.
----------------------------

Data type:  
   simple undirected graphs of order 0 to 68719476735.

Optional Header: 
   >>graph6<<     (without end of line!)

File name extension:
   .g6

One graph:
   Suppose G has n vertices.  Write the upper triangle of the adjacency
   matrix of G as a bit vector x of length n(n-1)/2, using the ordering
   (0,1),(0,2),(1,2),(0,3),(1,3),(2,3),...,(n-2,n-1).

   Then the graph is represented as  N(n) R(x).

Example:
   Suppose n=5 and G has edges 0-2, 0-4, 1-3 and 3-4.

   x = 0 10 010 1001
    
   Then N(n) = 68 and R(x) = R(010010 100100) = 81 99.
   So, the graph is  68 81 99.


Description of sparse6 format.
------------------------------

Data type:
   Undirected graphs of order 0 to 68719476735.
   Loops and multiple edges are permitted.
   (However, as of May 2022, the utilities in the nauty package
   and nauty itself do not support multiple edges and some
   utilities do not support loops either.)

Optional Header:
   >>sparse6<<     (without end of line!)

File name extension:
   .s6

General structure:

  Each graph occupies one text line. Except for the first character
  and end-of-line characters, each byte has the form 63+x, where 
  0 <= x <= 63. The byte encodes the six bits of x.

  The encoded graph consists of:
        (1) The character ':'.   (This is present to distinguish
                                  the code from graph6 format.)
        (2) The number of vertices.
        (3) A list of edges.
        (4) end-of-line

  Loops and multiple edges are supported, but not directed edges.

Number of vertices n:

  1, 4, or 8 bytes N(n) as above.
  This is the same as graph6 format.

List of edges:

  Let k be the number of bits needed to represent n-1 in binary.
  
  The remaining bytes encode a sequence 

      b[0] x[0] b[1] x[1] b[2] x[2] ... b[m] x[m]

  Each b[i] occupies 1 bit, and each x[i] occupies k bits.
  Pack them together in bigendian order, and pad up to a
  multiple of 6 as follows:
  1. If (n,k) = (2,1), (4,2), (8,3) or (16,4), and vertex
     n-2 has an edge but n-1 doesn't have an edge, and
     there are k+1 or more bits to pad, then pad with one
     0-bit and enough 1-bits to complete the multiple of 6.
  2. Otherwise, pad with enough 1-bits to complete the
     multiple of 6.
  These rules are to match the gtools procedures, and to avoid
  the padding from looking like an extra loop in unusual cases.

  Then represent this bit-stream 6 bits per byte as indicated above.

  The vertices of the graph are 0..n-1.
  The edges encoded by this sequence are determined thus:

     v = 0
     for i from 0 to m do
        if b[i] = 1 then v = v+1 endif;
        if x[i] > v then v = x[i] else output {x[i],v} endif
     endfor

   In decoding, an incomplete (b,x) pair at the end is discarded.

Example:

  :Fa@x^

  ':' indicates sparse6 format.
  Subtract 63 from the other bytes and write them in binary, 
  six bits each.

   000111 100010 000001 111001 011111

  The first byte is not 63, so it is n.  n=7
  n-1 needs 3 bits (k=3).  Write the other bits in groups
  of 1 and k:

    1 000  1 000  0 001  1 110   0 101  1 111
  
    This is the b/x sequence  1,0 1,0 0,1 1,6 0,5 1,7.
    The 1,7 at the end is just padding.
    The remaining parts give the edges 0-1 0-2 1-2 5-6.


Description of incremental sparse6 format.
-----------------------------------------

  This is an extension to sparse6 format that is very efficient if most
  graphs in a file are similar to the previous graph.

  Each graph occupies one text line. Except for the first character
  and end-of-line characters, each byte has the form 63+x, where 
  0 <= x <= 63. The byte encodes the six bits of x.

  The encoded graph consists of:
        (1) The character ';'.
        (2) A list of edges.
        (3) end-of-line

  This cannot appear as the first graph in a file.  The number of vertices
  is taken to be equal to the number of vertices in the previous graph.
  The list of edges specifies the symmetric difference of this graph and
  the previous graph.  It is encoded exactly the same as part (3) of
  sparse6 format.

  Loops are supported, but not multiple edges.


Description of digraph6 format.
------------------------------

Data type:  
   simple directed graphs (allowing loops) of order 0 to 68719476735.

Optional Header: 
   >>digraph6<<     (without end of line!)

File name extension:
   .d6

One graph:
   Suppose G has n vertices. Write the adjacency matrix of G
   as a bit vector x of length n^2, row by row.

   Then the graph is represented as '&' N(n) R(x).
   The character '&' (decimal 38) appears as the first character.

Example:
   Suppose n=5 and G has edges 0->2, 0->4, 3->1 and 3->4.

   x = 00101 00000 00000 01001 00000
    
   Then N(n) = 68 and
   R(x) = R(00101 00000 00000 01001 00000) = 73  63  65  79  63.
   So, the graph is  38 68 73  63  65  79  63.

Note:
   SageMath has a format also called "digraph6" that is different
   from this one. The leading "&" is omitted and the direction of
   every edge is reversed.
