# addEdges -- creates a list of graphs obtained by adding one new edge to the given graph in all possible ways

## Synopsis

• Usage:
Ll = addEdges L
Lg = addEdges G
Ls = addEdges S
• Inputs:
• L, a list, containing graphs in various formats
• S, , which describes a graph in Graph6 or Sparse6 format
• G, ,
• Optional inputs:
• MaxDegree => an integer, default value null, the maximum degree allowable for any vertex in the output graphs
• NoNew3Cycles => , default value false, whether graphs with new 3-cycles are allowed
• NoNew4Cycles => , default value false, whether graphs with new 4-cycles are allowed
• NoNew5Cycles => , default value false, whether graphs with new 5-cycles are allowed
• NoNewOddCycles => , default value false, whether graphs with new odd-cycles are allowed
• NoNewSmallCycles => an integer, default value null, an upper bound on cycles which are not allowed
• Outputs:
• Ll, a list, the list of graphs obtained from the graphs in $L$ in Graph6 format
• Lg, a list, the list of graphs obtained from $G$
• Ls, a list, the list of strings (in Graph6 format) obtained from $S$

## Description

Simply creates a list, in the same format as the input, of all possible graphs obtained by adding one new edge to the input graph.

 i1 : G = graph {{1,2},{2,3},{3,4},{4,5},{1,5}}; i2 : addEdges G o2 = {Graph{0 => {1, 2, 4}}, Graph{0 => {1, 3, 4}}, Graph{0 => {1, 4} }, 1 => {0, 2} 1 => {0, 2} 1 => {0, 2, 3} 2 => {1, 0, 3} 2 => {1, 3} 2 => {1, 3} 3 => {2, 4} 3 => {0, 2, 4} 3 => {1, 2, 4} 4 => {0, 3} 4 => {0, 3} 4 => {0, 3} ------------------------------------------------------------------------ Graph{0 => {1, 4} }, Graph{0 => {1, 4} }} 1 => {0, 2, 4} 1 => {0, 2} 2 => {1, 3} 2 => {1, 3, 4} 3 => {2, 4} 3 => {2, 4} 4 => {1, 0, 3} 4 => {0, 2, 3} o2 : List

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