WellTotallyDominated.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

Well totally dominated graphs

A total dominating set of G is a set S of vertices such that every vertex of G (including vertices in S) has a neighbor in S. A total dominating set S is minimal if no proper subset of S is also a total dominating set.

A graph G is well totally dominated if every minimal total dominating set has the same cardinality; equivalently, the total domination number equals the maximum cardinality of a minimal total dominating set.

The pendant vertices (also called leaves) of G are the vertices of degree 1.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def SimpleGraph.IsTotalDominatingSet A set S is a total dominating set of G if every vertex has a neighbor in S line 42 source
Source documentation
A set S is a total dominating set of G if every vertex has a neighbor in S.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, DecidableRel, SimpleGraph.Adj, Finset
Proof uses: Exists, And, Membership.mem
2 instances, coercions and auxiliary constants left out.
```
def IsTotalDominatingSet (G : SimpleGraph α) [DecidableRel G.Adj] (S : Finset α) : Prop :=
…
```
Mentions: SimpleGraph.S
def SimpleGraph.IsMinimalTotalDominatingSet A total dominating set S is minimal if no proper subset of S is also a total dominating set line 47 source
Source documentation
A total dominating set S is minimal if no proper subset of S is also a total dominating set.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, DecidableRel, SimpleGraph.Adj, Finset
Proof uses: And, SimpleGraph.IsTotalDominatingSet, LT.lt, Not
3 instances, coercions and auxiliary constants left out.
```
def IsMinimalTotalDominatingSet (G : SimpleGraph α) [DecidableRel G.Adj] (S : Finset α) : Prop :=
…
```
Mentions: SimpleGraph.S
def SimpleGraph.IsWellTotallyDominated A graph G is well totally dominated if every minimal total dominating set has the same cardinality line 53 source
Source documentation
A graph G is well totally dominated if every minimal total dominating set has the same cardinality.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, DecidableRel, SimpleGraph.Adj
Proof uses: Finset, SimpleGraph.IsMinimalTotalDominatingSet, Eq, Nat, Finset.card
```
def IsWellTotallyDominated (G : SimpleGraph α) [DecidableRel G.Adj] : Prop :=
…
```
def SimpleGraph.pendantVertices The set of pendant vertices (leaves) of G: vertices of degree 1 line 60 source
Source documentation
The set of pendant vertices (leaves) of G: vertices of degree 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, Finset
Proof uses: Finset.filter, Eq, Nat, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, OfNat.ofNat, Finset.univ
5 instances, coercions and auxiliary constants left out.
```
noncomputable def pendantVertices (G : SimpleGraph α) [DecidableRel G.Adj] : Finset α :=
…
```