GraphVertexConnectivity
The fewest vertices whose deletion leaves a disconnected graph; the complete graph on n vertices gives n - 1.
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St000262via LabeledGraph✓GraphVertexConnectivity(graph)The fewest vertices whose deletion leaves a disconnected graph; the complete graph on n vertices gives n - 1.Typed signatures
| Type | Package | Applies |
|---|---|---|
(labeled_graph) -> number | combinatorics | always |
Details
- Defined over
LabeledGraphas an expression in_x, evaluated by compute-engine — the definition IS the implementation. - Takes a
LabeledGraphand nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer. - FindStat's St000262. A disconnected graph gives 0. The graph with no vertices gives 0.
Definition
Epsil, as the implementation runs it.
on LabeledGraph, over _x
0 if _x[1] == 0 else Apply((n: integer) => Apply((edges: list<list<integer>>) => Apply(
(inside: list<integer>) => Apply(
(totals: list<integer>) => Apply(
(connected: list<integer>) => Min(Map(
Function(totals[m + 1] if (1 if m <
2 ^ n -
1 else 0) *
(1 -
connected[2 ^ n -
1 -
m +
1]) ==
1 else n - 1, m),
Range(0, 2 ^ n -
1),
)),
Apply((a_init: list<integer>) => Fold((a: list<integer>, b) => Apply(
(a_next: list<integer>) => a_next,
Join(a, [1 if totals[b + 1] ==
1 else Min(1, Max(Sum(Map(Function(a[b -
2 ^
(v - 1) +
1] *
(1 if inside[b + 1] -
inside[b -
2 ^
(v - 1) +
1] >
0 else 0) if b %
2 ^ v >=
2 ^
(v - 1) else 0, v), Range(1, n))), 0))]),
), a_init, Range(1, 2 ^ n -
1)), [0]),
),
Map(
Function(Max(Sum(Map(Function(1 if m %
2 ^ v >=
2 ^
(v - 1) else 0, v), Range(1, n))), 0), m),
Range(0, 2 ^ n -
1),
),
),
Map(
Function(Max(Sum(Map(Function((1 if m %
2 ^ e[1] >=
2 ^
(e[1] - 1) else 0) *
(1 if m %
2 ^ e[2] >=
2 ^
(e[2] - 1) else 0), e), edges)), 0), m),
Range(0, 2 ^ n -
1),
),
), _x[2]), _x[1])Examples
Properties #
About
| Examples | 4 |
|---|---|
| Crosswalk | FindStat |