# buildGraphFilter -- creates the appropriate filter string for use with filterGraphs and countGraphs

## Synopsis

• Usage:
s = buildGraphFilter h
s = buildGraphFilter l
• Inputs:
• h, , which describes the properties desired in filtering
• l, a list, which describes the properties desired in filtering
• Outputs:

## Description

The filterGraphs and countGraphs methods both can use a tremendous number of constraints which are described by a rather tersely encoded string. This method builds that string given information in the HashTable h or the List l. Any keys which do not exist are simply ignored and any values which are not valid (e.g., exactly -3 vertices) are also ignored.

The values can either be Boolean or in ZZ. Boolean values are treated exactly as expected. Numerical values are more complicated; we use an example to illustrate how numerical values can be used, but note that this usage works for all numerically valued keys.

The key NumEdges restricts to a specific number of edges in the graph. If the value is the integer n, then only graphs with exactly n edges are returned.

 `i1 : L = {graph {{1,2}}, graph {{1,2},{2,3}}, graph {{1,2},{2,3},{3,4}}, graph {{1,2},{2,3},{3,4},{4,5}}};` `i2 : s = buildGraphFilter {"NumEdges" => 3};` ```i3 : filterGraphs(L, s) o3 = {Graph{1 => {2} }} 2 => {1, 3} 3 => {2, 4} 4 => {3} o3 : List```

If the value is the Sequence (m,n), then all graphs with at least m and at most n edges are returned.

 `i4 : s = buildGraphFilter {"NumEdges" => (2,3)};` ```i5 : filterGraphs(L, s) o5 = {Graph{1 => {2} }, Graph{1 => {2} }} 2 => {1, 3} 2 => {1, 3} 3 => {2} 3 => {2, 4} 4 => {3} o5 : List```

If the value is the Sequence (,n), then all graphs with at most n edges are returned.

 `i6 : s = buildGraphFilter {"NumEdges" => (,3)};` ```i7 : filterGraphs(L, s) o7 = {Graph{1 => {2}}, Graph{1 => {2} }, Graph{1 => {2} }} 2 => {1} 2 => {1, 3} 2 => {1, 3} 3 => {2} 3 => {2, 4} 4 => {3} o7 : List```

If the value is the Sequence (m,), then all graphs with at least m edges are returned.

 `i8 : s = buildGraphFilter {"NumEdges" => (2,)};` ```i9 : filterGraphs(L, s) o9 = {Graph{1 => {2} }, Graph{1 => {2} }, Graph{1 => {2} }} 2 => {1, 3} 2 => {1, 3} 2 => {1, 3} 3 => {2} 3 => {2, 4} 3 => {2, 4} 4 => {3} 4 => {3, 5} 5 => {4} o9 : List```

Moreover, the associated key NegateNumEdges, if true, causes the opposite to occur.

 `i10 : s = buildGraphFilter {"NumEdges" => (2,), "NegateNumEdges" => true};` ```i11 : filterGraphs(L, s) o11 = {Graph{1 => {2}}} 2 => {1} o11 : List```

The following are the boolean options: "Regular", "Bipartite", "Eulerian", "VertexTransitive".

The following are the numerical options (recall all have the associate "Negate" option): "NumVertices", "NumEdges", "MinDegree", "MaxDegree", "Radius", "Diameter", "Girth", "NumCycles", "NumTriangles", "GroupSize", "Orbits", "FixedPoints", "Connectivity", "MinCommonNbrsAdj", "MaxCommonNbrsAdj", "MinCommonNbrsNonAdj", "MaxCommonNbrsNonAdj".

## Caveat

Connectivity only works for the values 0, 1, 2 and uses the following definition of k-connectivity. A graph is k-connected if k is the minimum size of a set of vertices whose complement is not connected.

Thus, in order to filter for connected graphs, one must use {"Connectivity" => 0, "NegateConnectivity" => true}.

NumCycles can only be used with graphs on at most n vertices, where n is the number of bits for which nauty was compiled, typically 32 or 64.