# neighborhoodComplements -- complements the neighborhood for each vertex, individually

## Synopsis

• Usage:
N' = neighborhoodComplements L
N = neighborhoodComplements S
N = neighborhoodComplements G
• Inputs:
• L, a list, containing graphs in various formats
• S, , a graph encoded in either Sparse6 or Graph6 format
• G, ,
• Outputs:
• N', a list, containing graphs in either Graph6 or Sparse6 format, modified as described below
• N, a list, containing graphs, in the same format as the input, modified as described below

## Description

The method creates a list of graphs, one for each vertex of the original graph $G$. The graph associated to a vertex $v$ of $G$ has the neighborhood of $v$ complemented.

The method does not remove isomorphs.

 i1 : R = QQ[a..e]; i2 : neighborhoodComplements graph {a*b, a*c, b*c, c*d, d*e} o2 = {Graph{edges => {{a, b}, {a, c}, {b, c}, {b, d}, {b, e}, {c, e}, {d, ring => R vertices => {a, b, c, d, e} ------------------------------------------------------------------------ e}}}, Graph{edges => {{a, b}, {a, c}, {b, c}, {a, d}, {a, e}, {c, e}, ring => R vertices => {a, b, c, d, e} ------------------------------------------------------------------------ {d, e}}}, Graph{edges => {{a, b}, {a, c}, {b, c}, {c, d}, {a, e}, {b, ring => R vertices => {a, b, c, d, e} ------------------------------------------------------------------------ e}}}, Graph{edges => {{a, b}, {c, d}, {a, e}, {b, e}, {d, e}}}, ring => R vertices => {a, b, c, d, e} ------------------------------------------------------------------------ Graph{edges => {{a, b}, {a, c}, {b, c}, {a, d}, {b, d}, {d, e}}}} ring => R vertices => {a, b, c, d, e} o2 : List