Introduction DI-Pathological Graphs
Some Bounds on DI-Pathological Graphs
John Asplund, Joe Chaffee, and James M. Hammer∗
Department of Mathematics and StatisticsAuburn University
July 9, 2014
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Introduction DI-Pathological Graphs
Unrelated Problem
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Introduction DI-Pathological Graphs
Unrelated Problem
Hidato
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Introduction DI-Pathological Graphs
Unrelated Problem
Natural Questions
What is the least amount of numbers that need to beprescribed to have a unique solution?
Are there any forbidden prescriptions?
If there are forbidden presctiptions, how big does a squareneed to be to embedd the smaller one?
If we change the board shape, what can we say about thispuzzle?
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Introduction DI-Pathological Graphs
Unrelated Problem
Natural Questions
What is the least amount of numbers that need to beprescribed to have a unique solution?
Are there any forbidden prescriptions?
If there are forbidden presctiptions, how big does a squareneed to be to embedd the smaller one?
If we change the board shape, what can we say about thispuzzle?
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Introduction DI-Pathological Graphs
Unrelated Problem
Natural Questions
What is the least amount of numbers that need to beprescribed to have a unique solution?
Are there any forbidden prescriptions?
If there are forbidden presctiptions, how big does a squareneed to be to embedd the smaller one?
If we change the board shape, what can we say about thispuzzle?
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Introduction DI-Pathological Graphs
Unrelated Problem
Natural Questions
What is the least amount of numbers that need to beprescribed to have a unique solution?
Are there any forbidden prescriptions?
If there are forbidden presctiptions, how big does a squareneed to be to embedd the smaller one?
If we change the board shape, what can we say about thispuzzle?
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Introduction DI-Pathological Graphs
Applications
Why Do We Care?
Graph Domination can be used in the following areas:
Design & Analysis of Communication Networks
Social Sciences
Optimization
Bio-informatics
Computational Complexity
Algorithm Design
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Introduction DI-Pathological Graphs
Applications
Why Do We Care?
Graph Domination can be used in the following areas:
Design & Analysis of Communication Networks
Social Sciences
Optimization
Bio-informatics
Computational Complexity
Algorithm Design
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Introduction DI-Pathological Graphs
Applications
Why Do We Care?
Graph Domination can be used in the following areas:
Design & Analysis of Communication Networks
Social Sciences
Optimization
Bio-informatics
Computational Complexity
Algorithm Design
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Introduction DI-Pathological Graphs
Applications
Why Do We Care?
Graph Domination can be used in the following areas:
Design & Analysis of Communication Networks
Social Sciences
Optimization
Bio-informatics
Computational Complexity
Algorithm Design
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Introduction DI-Pathological Graphs
Applications
Why Do We Care?
Graph Domination can be used in the following areas:
Design & Analysis of Communication Networks
Social Sciences
Optimization
Bio-informatics
Computational Complexity
Algorithm Design
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Introduction DI-Pathological Graphs
Applications
Why Do We Care?
Graph Domination can be used in the following areas:
Design & Analysis of Communication Networks
Social Sciences
Optimization
Bio-informatics
Computational Complexity
Algorithm Design
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Dominating Set)
A set D ⊂ V is a dominating set of a graph G = (V,E) ifeach vertex in V is either in D or is adjacent to a vertex in D.
Example
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Dominating Set)
A set D ⊂ V is a dominating set of a graph G = (V,E) ifeach vertex in V is either in D or is adjacent to a vertex in D.
Example
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Dominating Set)
A set D ⊂ V is a dominating set of a graph G = (V,E) ifeach vertex in V is either in D or is adjacent to a vertex in D.
Example
Figure: Dominating Set
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Domination Number)
The domination number, γ(G) is the minimum cardinality ofa dominating set of G.
Definition (Minimum Dominating Set)
A minimum dominating set of a graph G is a dominatingset of size γ(G).
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Domination Number)
The domination number, γ(G) is the minimum cardinality ofa dominating set of G.
Definition (Minimum Dominating Set)
A minimum dominating set of a graph G is a dominatingset of size γ(G).
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Independent Set)
A set I ⊂ V is a independent set of a graph G = (V,E) if notwo vertices in I are adjacent.
Example
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Independent Set)
A set I ⊂ V is a independent set of a graph G = (V,E) if notwo vertices in I are adjacent.
Example
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Independent Set)
A set I ⊂ V is a independent set of a graph G = (V,E) if notwo vertices in I are adjacent.
Example
Figure: Independent Set
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Independence Number)
The independence number, α(G) is the maximumcardinality of an independent set of G.
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Inverse Dominating Set)
Let D be a minimum dominating set. A set S ⊂ V \D is aninverse dominating set of a graph G = (V,E) with noisolated vertices if each vertex in V is either in S or is adjacentto a vertex in S.
Example
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Inverse Dominating Set)
Let D be a minimum dominating set. A set S ⊂ V \D is aninverse dominating set of a graph G = (V,E) with noisolated vertices if each vertex in V is either in S or is adjacentto a vertex in S.
Example
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Inverse Dominating Set)
Let D be a minimum dominating set. A set S ⊂ V \D is aninverse dominating set of a graph G = (V,E) with noisolated vertices if each vertex in V is either in S or is adjacentto a vertex in S.
Example
Figure: Dominating Set
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Inverse Dominating Set)
Let D be a minimum dominating set. A set S ⊂ V \D is aninverse dominating set of a graph G = (V,E) with noisolated vertices if each vertex in V is either in S or is adjacentto a vertex in S.
Example
Figure: Dominating Set vs. Inverse Dominating Set
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Inverse Domination Number)
The inverse domination number, γ′(G) is the minimumcardinality of any inverse dominating set over all minimumdominating sets.
Conjecture. (Hedetniemi)
γ′(G) ≤ α(G) for all graphs G.
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Introduction DI-Pathological Graphs
Definition Theory
Definition (Inverse Domination Number)
The inverse domination number, γ′(G) is the minimumcardinality of any inverse dominating set over all minimumdominating sets.
Conjecture. (Hedetniemi)
γ′(G) ≤ α(G) for all graphs G.
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Introduction DI-Pathological Graphs
Definition
Definition (Maximal Independent Set)
An independent set I is said to be a maximal independentset of a graph G if I dominates G.
Example
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Introduction DI-Pathological Graphs
Definition
Definition (Maximal Independent Set)
An independent set I is said to be a maximal independentset of a graph G if I dominates G.
Example
Figure: Maximal Independent Set
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Introduction DI-Pathological Graphs
Preliminary Observations
True Fact.
If G has a minimum dominating set D that is disjoint from amaximal independent set I, then γ′(G) ≤ α(G).
Proof.
γ′(G) ≤ |I| ≤ α(G).
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Introduction DI-Pathological Graphs
Preliminary Observations
True Fact.
If G has a minimum dominating set D that is disjoint from amaximal independent set I, then γ′(G) ≤ α(G).
Proof.
γ′(G) ≤ |I| ≤ α(G).
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Introduction DI-Pathological Graphs
Preliminary Observations
Observation
If there is a counterexample to Hedetniemi’s conjecture, it mustbe a graph where every maximal independent set intersectsevery minimum dominating set.
Definition (DI-Pathological Graph)
A graph G = (V,E) is said to be DI-pathological if everymaximal independent set intersects every minimum dominatingset.
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Introduction DI-Pathological Graphs
Preliminary Observations
Observation
If there is a counterexample to Hedetniemi’s conjecture, it mustbe a graph where every maximal independent set intersectsevery minimum dominating set.
Definition (DI-Pathological Graph)
A graph G = (V,E) is said to be DI-pathological if everymaximal independent set intersects every minimum dominatingset.
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Introduction DI-Pathological Graphs
Preliminary Observations
Natural Characterization Question
How small can a DI-pathological graph be with respect to agiven domination number?
Two Ways to Minimize
The number of vertices
The number of Edges
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Introduction DI-Pathological Graphs
Preliminary Observations
Natural Characterization Question
How small can a DI-pathological graph be with respect to agiven domination number?
Two Ways to Minimize
The number of vertices
The number of Edges
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Introduction DI-Pathological Graphs
Preliminary Observations
Natural Characterization Question
How small can a DI-pathological graph be with respect to agiven domination number?
Two Ways to Minimize
The number of vertices
The number of Edges
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Introduction DI-Pathological Graphs
History
True Fact. (Prier)
There does not exist a DI-pathological graph with γ(G) = 1other than K1.
Theorem (Prier)
If G is a connected DI-pathological graph with γ(G) = 2, thenG = Km,n where m,n ≥ 3.
Smallest Graph with γ(G) = 2
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Introduction DI-Pathological Graphs
History
True Fact. (Prier)
There does not exist a DI-pathological graph with γ(G) = 1other than K1.
Theorem (Prier)
If G is a connected DI-pathological graph with γ(G) = 2, thenG = Km,n where m,n ≥ 3.
Smallest Graph with γ(G) = 2
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Introduction DI-Pathological Graphs
History
True Fact. (Prier)
There does not exist a DI-pathological graph with γ(G) = 1other than K1.
Theorem (Prier)
If G is a connected DI-pathological graph with γ(G) = 2, thenG = Km,n where m,n ≥ 3.
Smallest Graph with γ(G) = 2
Figure: Minimum Dominating Set15 / 25
Introduction DI-Pathological Graphs
History
True Fact. (Prier)
There does not exist a DI-pathological graph with γ(G) = 1other than K1.
Theorem (Prier)
If G is a connected DI-pathological graph with γ(G) = 2, thenG = Km,n where m,n ≥ 3.
Smallest Graph with γ(G) = 2
Figure: Maximal Independent Set15 / 25
Introduction DI-Pathological Graphs
History
True Fact. (Prier)
There does not exist a DI-pathological graph with γ(G) = 1other than K1.
Theorem (Prier)
If G is a connected DI-pathological graph with γ(G) = 2, thenG = Km,n where m,n ≥ 3.
Smallest Graph with γ(G) = 2
Figure: K3,3
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Introduction DI-Pathological Graphs
History
Theorem (Prier)
If G is a connected graph with γ(G) = 3, then the smallestDI-pathological graph (with respect to both vertices and edges)has 9 vertices and 10 edges.
The Smallest connected DI-pathological graph with γ(G) = 3
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Introduction DI-Pathological Graphs
History
Theorem (Prier)
If G is a connected graph with γ(G) = 3, then the smallestDI-pathological graph (with respect to both vertices and edges)has 9 vertices and 10 edges.
The Smallest connected DI-pathological graph with γ(G) = 3
Figure: The Prier Graph
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Introduction DI-Pathological Graphs
Conjecture
False Fact.
The unique connected, DI-pathological graph G with the fewestnumber of edges and the fewest number of vertices for γ(G) ≥ 3is two 4-cycles connected by a path of length 3γ(G)− 7.
So, |V (G)| = 3γ(G), |E| = 3γ(G) + 1
Proposed Smallest Graph
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Introduction DI-Pathological Graphs
Conjecture
False Fact.
The unique connected, DI-pathological graph G with the fewestnumber of edges and the fewest number of vertices for γ(G) ≥ 3is two 4-cycles connected by a path of length 3γ(G)− 7.
So, |V (G)| = 3γ(G), |E| = 3γ(G) + 1
Proposed Smallest Graph
︸ ︷︷ ︸P3γ(G)−7
Figure: γ(G) ≥ 3
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Introduction DI-Pathological Graphs
9 Vertices, γ(G) = 3
Classification
Figure: DI-pathological graphs on 9 vertices with γ(G) = 3
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Introduction DI-Pathological Graphs
Smallest DI-pathological Graph (Vertices)
Theorem
Let G be a connected, DI-pathological graph with dominationnumber γ(G) where γ(G) ≥ 4. Then |V (G)| ≥ 2γ(G) + 4.
Construction
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Introduction DI-Pathological Graphs
Smallest DI-pathological Graph (Vertices)
Theorem
Let G be a connected, DI-pathological graph with dominationnumber γ(G) where γ(G) ≥ 4. Then |V (G)| ≥ 2γ(G) + 4.
Construction
︷ ︸︸ ︷2γ − 8 vertices
Figure: Smallest with Respect to Vertices
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Introduction DI-Pathological Graphs
Smallest DI-pathological Graph (Vertices)
Theorem
Let G be a connected, DI-pathological graph with dominationnumber γ(G) where γ(G) ≥ 4. Then |V (G)| ≥ 2γ(G) + 4.
Construction
︷ ︸︸ ︷2γ − 8 vertices
Figure: Smallest with Respect to Vertices
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Introduction DI-Pathological Graphs
Smallest DI-pathological Graph (Vertices)
Theorem
Let G be a connected, DI-pathological graph with dominationnumber γ(G) where γ(G) ≥ 4. Then |V (G)| ≥ 2γ(G) + 4.
Construction
︷ ︸︸ ︷2γ − 6 vertices
Figure: Another Smallest with Respect to Vertices
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Introduction DI-Pathological Graphs
Smallest DI-pathological Graph (Vertices)
Theorem
Let G be a connected, DI-pathological graph with dominationnumber γ(G) where γ(G) ≥ 4. Then |V (G)| ≥ 2γ(G) + 4.
Construction
︷ ︸︸ ︷2γ − 6 vertices
Figure: Another Smallest with Respect to Vertices
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Introduction DI-Pathological Graphs
Notes on Vertex Minimal
Observation
If γ(G) = 4 then 3γ(G) = 2γ(G) + 4.
However, for γ ≥ 5, 2γ + 4 < 3γ.
Theorem (Prier)
If G is a connected graph such that γ(G) ≤ 4 thenγ′(G) ≤ α(G).
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Introduction DI-Pathological Graphs
Notes on Vertex Minimal
Observation
If γ(G) = 4 then 3γ(G) = 2γ(G) + 4.
However, for γ ≥ 5, 2γ + 4 < 3γ.
Theorem (Prier)
If G is a connected graph such that γ(G) ≤ 4 thenγ′(G) ≤ α(G).
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Introduction DI-Pathological Graphs
Notes on Vertex Minimal
Observation
If γ(G) = 4 then 3γ(G) = 2γ(G) + 4.
However, for γ ≥ 5, 2γ + 4 < 3γ.
Theorem (Prier)
If G is a connected graph such that γ(G) ≤ 4 thenγ′(G) ≤ α(G).
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Introduction DI-Pathological Graphs
Smallest DI-pathological Graph (Edges)
Theorem
Let G be a connected, DI-pathological graph with dominationnumber γ(G) where γ(G) ≥ 4. Then |E(G)| ≥ 2γ(G) + 5.
Construction
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Introduction DI-Pathological Graphs
Smallest DI-pathological Graph (Edges)
Theorem
Let G be a connected, DI-pathological graph with dominationnumber γ(G) where γ(G) ≥ 4. Then |E(G)| ≥ 2γ(G) + 5.
Construction
︷ ︸︸ ︷2γ − 8 vertices
Figure: Smallest with Respect to Vertices
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Introduction DI-Pathological Graphs
Smallest DI-pathological Graph (Edges)
Theorem
Let G be a connected, DI-pathological graph with dominationnumber γ(G) where γ(G) ≥ 4. Then |E(G)| ≥ 2γ(G) + 5.
Construction
︷ ︸︸ ︷2γ − 8 vertices
Figure: Smallest with Respect to Vertices
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Introduction DI-Pathological Graphs
Smallest DI-pathological Graph (Edges)
Theorem
Let G be a connected, DI-pathological graph with dominationnumber γ(G) where γ(G) ≥ 4. Then |E(G)| ≥ 2γ(G) + 5.
Construction
︷ ︸︸ ︷γ − 4
Figure: Another Smallest with Respect to Vertices
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Introduction DI-Pathological Graphs
Smallest DI-pathological Graph (Edges)
Theorem
Let G be a connected, DI-pathological graph with dominationnumber γ(G) where γ(G) ≥ 4. Then |E(G)| ≥ 2γ(G) + 5.
Construction
︷ ︸︸ ︷γ − 4
Figure: Another Smallest with Respect to Vertices
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Introduction DI-Pathological Graphs
Notes on Edge Minimal
Observation
If γ(G) = 4 then 3γ(G) + 1 = 2γ(G) + 5.
However, for γ ≥ 5, 2γ + 5 < 3γ + 1.
Observation
It was (secretly) our hope that one of these would be acounterexample to Hedetniemi’s conjecture; however, none ofthese graphs disprove Hedetniemi’s conjecture.
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Introduction DI-Pathological Graphs
Notes on Edge Minimal
Observation
If γ(G) = 4 then 3γ(G) + 1 = 2γ(G) + 5.
However, for γ ≥ 5, 2γ + 5 < 3γ + 1.
Observation
It was (secretly) our hope that one of these would be acounterexample to Hedetniemi’s conjecture; however, none ofthese graphs disprove Hedetniemi’s conjecture.
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Introduction DI-Pathological Graphs
Notes on Edge Minimal
Observation
If γ(G) = 4 then 3γ(G) + 1 = 2γ(G) + 5.
However, for γ ≥ 5, 2γ + 5 < 3γ + 1.
Observation
It was (secretly) our hope that one of these would be acounterexample to Hedetniemi’s conjecture; however, none ofthese graphs disprove Hedetniemi’s conjecture.
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Introduction DI-Pathological Graphs
Disconnected Case
Theorem
Let G be a DI-pathological graph with no isolated vertices anddomination number γ(G). Then |V (G)| ≥ 2γ(G) + 2 and|E(G)| ≥ γ(G) + 7.
Example
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Introduction DI-Pathological Graphs
Disconnected Case
Theorem
Let G be a DI-pathological graph with no isolated vertices anddomination number γ(G). Then |V (G)| ≥ 2γ(G) + 2 and|E(G)| ≥ γ(G) + 7.
Example
Figure: Smallest DI-pathological graph with no isolated vertices.
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Introduction DI-Pathological Graphs
Disconnected Case
Theorem
Let G be a DI-pathological graph with no isolated vertices anddomination number γ(G). Then |V (G)| ≥ 2γ(G) + 2 and|E(G)| ≥ γ(G) + 7.
Example
Figure: Smallest DI-pathological graph with no isolated vertices.
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Introduction DI-Pathological Graphs
Open Problems
Things that make you go hmmmmm. . .
A Counterexample or a proof of Hedetniemi’s Conjecture
Characterize all of the graphs with the lowest number ofvertices for γ = 5
Characterize all of the graphs with the lowest number ofedges for γ = 5
The intersection of these two characterizations is a uniquegraph, namely:
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Introduction DI-Pathological Graphs
Open Problems
Things that make you go hmmmmm. . .
A Counterexample or a proof of Hedetniemi’s Conjecture
Characterize all of the graphs with the lowest number ofvertices for γ = 5
Characterize all of the graphs with the lowest number ofedges for γ = 5
The intersection of these two characterizations is a uniquegraph, namely:
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Introduction DI-Pathological Graphs
Open Problems
Things that make you go hmmmmm. . .
A Counterexample or a proof of Hedetniemi’s Conjecture
Characterize all of the graphs with the lowest number ofvertices for γ = 5
Characterize all of the graphs with the lowest number ofedges for γ = 5
The intersection of these two characterizations is a uniquegraph, namely:
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Introduction DI-Pathological Graphs
Open Problems
Things that make you go hmmmmm. . .
A Counterexample or a proof of Hedetniemi’s Conjecture
Characterize all of the graphs with the lowest number ofvertices for γ = 5
Characterize all of the graphs with the lowest number ofedges for γ = 5
The intersection of these two characterizations is a uniquegraph, namely:
︷ ︸︸ ︷2γ − 8 vertices
Figure: Proposed Intersection
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Introduction DI-Pathological Graphs
War Eagle!
Thank You For YourKind Attention !
Figure: War Eagle !
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