newEdges -- replaces disjoint pairs of edges by disjoint pairs of two-chains

Synopsis

Usage:

N' = newEdges L

N = newEdges S

N = newEdges(G, R)
Inputs:
- L, a list, containing graphs in various formats
- S, a string, a graph encoded in either Sparse6 or Graph6 format
- G, a graph,
- R, a polynomial ring, a ring with exactly two more variables than the ring of $G$
Outputs:
- N', a list, containing graphs in either Graph6 or Sparse6 format, modified as described below
- N, a list, a list of graphs, in the same format as the input, modified as described below

Description

The list of "new edge" graphs are formed as follows: Let $ab$ and $cd$ be disjoint edges of $G$. Then the associated "new edge" graph $H$ is $G$ with the edges $ab$ and $cd$ removed, the vertices $e$ and $f$ added, and the new edges $ae, be, cf, df,$ and $ef$ added.

i1 : R = QQ[a..d];

i2 : G = graph {a*b, c*d};

i3 : S = QQ[a..f];

i4 : newEdges(G, S)

o4 = {Graph{"edges" => {{a, e}, {b, e}, {c, f}, {d, f}, {e, f}}}}
            "ring" => S
            "vertices" => {a, b, c, d, e, f}

o4 : List

If the List input format is used, then one should use care as the list may contain isomorphic pairs.

Ways to use newEdges :

newEdges(Graph,PolynomialRing)
newEdges(List)
newEdges(String)

For the programmer

The object newEdges is a method function.