Some Bounds on DI-Pathological Graphs€¦ · IntroductionDI-Pathological Graphs Some Bounds on...

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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Unrelated Problem

Hidato

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31 1 4 46

34 36 40

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Unrelated Problem

Hidato

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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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Unrelated Problem

Natural Questions

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Unrelated Problem

Natural Questions

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Unrelated Problem

Natural Questions

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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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Applications

Why Do We Care?

Social Sciences

Optimization

Bio-informatics

Algorithm Design

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Applications

Why Do We Care?

Social Sciences

Optimization

Bio-informatics

Algorithm Design

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Applications

Why Do We Care?

Social Sciences

Optimization

Bio-informatics

Algorithm Design

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Applications

Why Do We Care?

Social Sciences

Optimization

Bio-informatics

Algorithm Design

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Applications

Why Do We Care?

Social Sciences

Optimization

Bio-informatics

Algorithm Design

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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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Definition Theory

Example

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Definition Theory

Example

Figure: Dominating Set

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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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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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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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Definition Theory

Example

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Definition Theory

Example

Figure: Independent Set

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Definition Theory

Definition (Independence Number)

The independence number, α(G) is the maximumcardinality of an independent set of G.

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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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Definition Theory

Example

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Definition Theory

Example

Figure: Dominating Set

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Definition Theory

Example

Figure: Dominating Set vs. Inverse Dominating Set

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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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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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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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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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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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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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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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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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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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The number of Edges

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The number of Edges

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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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History

True Fact. (Prier)

Theorem (Prier)

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History

True Fact. (Prier)

Theorem (Prier)

Figure: Minimum Dominating Set15 / 25

History

True Fact. (Prier)

Theorem (Prier)

Figure: Maximal Independent Set15 / 25

History

True Fact. (Prier)

Theorem (Prier)

Figure: K3,3

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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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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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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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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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9 Vertices, γ(G) = 3

Classification

Figure: DI-pathological graphs on 9 vertices with γ(G) = 3

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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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Theorem

Construction

︷︸︸︷2γ − 8 vertices

Figure: Smallest with Respect to Vertices

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Theorem

Construction

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Theorem

Construction

Figure: Another Smallest with Respect to Vertices

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Theorem

Construction

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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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Observation

If γ(G) = 4 then 3γ(G) = 2γ(G) + 4.

Theorem (Prier)

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Observation

If γ(G) = 4 then 3γ(G) = 2γ(G) + 4.

Theorem (Prier)

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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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Theorem

Construction

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Theorem

Construction

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Theorem

Construction

︷︸︸︷γ − 4

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Theorem

Construction

︷︸︸︷γ − 4

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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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Observation

If γ(G) = 4 then 3γ(G) + 1 = 2γ(G) + 5.

However, for γ ≥ 5, 2γ + 5 < 3γ + 1.

Observation

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Observation

If γ(G) = 4 then 3γ(G) + 1 = 2γ(G) + 5.

However, for γ ≥ 5, 2γ + 5 < 3γ + 1.

Observation

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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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Disconnected Case

Theorem

Example

Figure: Smallest DI-pathological graph with no isolated vertices.

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Disconnected Case

Theorem

Example

Figure: Smallest DI-pathological graph with no isolated vertices.

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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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Open Problems

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Open Problems

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Open Problems

Figure: Proposed Intersection

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War Eagle!

Thank You For YourKind Attention !

Figure: War Eagle !

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