Intersection Graphs of Maximal Convex Sub-Polygons of k -Lizards - - PowerPoint PPT Presentation

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Intersection Graphs of Maximal Convex Sub-Polygons of k -Lizards - - PowerPoint PPT Presentation

Feb 04, 2024 131 likes •1.5k views

Geometric Representations of Graphs Intersection Graphs of Maximal Convex Sub-Polygons of k -Lizards Caroline Daugherty Josh Laison Rebecca Robinson Kyle Salois Willamette Mathematics Consortium REU Willamette REU k -MSP Intersection Graphs

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Geometric Representations of Graphs

Intersection Graphs of Maximal Convex Sub-Polygons of k-Lizards

Caroline Daugherty Josh Laison Rebecca Robinson Kyle Salois

Willamette Mathematics Consortium REU

Willamette REU k-MSP Intersection Graphs August 4, 2017 1 / 67

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Outline

1

Introduction Intersection Graphs Definitions Useful Lemmas

2

Results Graphs that are k-MSP Graphs Separating Examples Generalizations and Conjectures

3

Future Research

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Intersection Graphs A graph G is a set V(G) of points called vertices, alongside a set of edges, which connect the vertices and form a set E(G). An intersection graph H is a graph formed by a family of sets where each vertex represents a set, and two vertices have an edge between them whenever the corresponding sets intersect.

A B C D E a b c d e Willamette REU k-MSP Intersection Graphs August 4, 2017 3 / 67

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Maximal Rectangles of a Polyomino: Shearer 1982, Maire 1993 A polyomino is a geometric figure formed by joining unit squares in the plane.

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Maximal Rectangles of a Polyomino: Shearer 1982, Maire 1993 A polyomino is a geometric figure formed by joining unit squares in the plane. We can draw an intersection graph by creating a vertex for every maximal rectangle and connecting two vertices with an edge if their maximal rectangles intersect.

Willamette REU k-MSP Intersection Graphs August 4, 2017 4 / 67

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Maximal Rectangles of a Polyomino: Shearer 1982, Maire 1993 A polyomino is a geometric figure formed by joining unit squares in the plane. We can draw an intersection graph by creating a vertex for every maximal rectangle and connecting two vertices with an edge if their maximal rectangles intersect.

Willamette REU k-MSP Intersection Graphs August 4, 2017 4 / 67

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Maximal Rectangles of a Polyomino: Shearer 1982, Maire 1993 A polyomino is a geometric figure formed by joining unit squares in the plane. We can draw an intersection graph by creating a vertex for every maximal rectangle and connecting two vertices with an edge if their maximal rectangles intersect.

Willamette REU k-MSP Intersection Graphs August 4, 2017 4 / 67

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Maximal Rectangles of a Polyomino: Shearer 1982, Maire 1993 A polyomino is a geometric figure formed by joining unit squares in the plane. We can draw an intersection graph by creating a vertex for every maximal rectangle and connecting two vertices with an edge if their maximal rectangles intersect.

Willamette REU k-MSP Intersection Graphs August 4, 2017 4 / 67

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Maximal Rectangles of a Polyomino: Shearer 1982, Maire 1993 A polyomino is a geometric figure formed by joining unit squares in the plane. We can draw an intersection graph by creating a vertex for every maximal rectangle and connecting two vertices with an edge if their maximal rectangles intersect.

Willamette REU k-MSP Intersection Graphs August 4, 2017 4 / 67

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Maximal Rectangles of a Polyomino Shearer 1982, Maire 1993 All intersection graphs formed by the maximal rectangles of a polyomino are perfect.

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Maximal Rectangles of a Polyomino Shearer 1982, Maire 1993 All intersection graphs formed by the maximal rectangles of a polyomino are perfect. In a perfect graph, the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph.

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Maximal Rectangles of a Polyomino Shearer 1982, Maire 1993 All intersection graphs formed by the maximal rectangles of a polyomino are perfect. In a perfect graph, the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. An induced subgraph is a set of vertices of a graph G that are connected by all the same edges that connect them in G.

Willamette REU k-MSP Intersection Graphs August 4, 2017 5 / 67

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Maximal Rectangles of a Polyomino Shearer 1982, Maire 1993 All intersection graphs formed by the maximal rectangles of a polyomino are perfect. In a perfect graph, the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. An induced subgraph is a set of vertices of a graph G that are connected by all the same edges that connect them in G. A clique in a graph is a set of vertices where every pair has an edge.

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Definitions A k-lizard is a simply connected polygon such that each side has slope in the set {i⇡/k, i⇡/k, 0 | 1  i  k}.

An example of a 4-lizard.

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Definitions The direction of a side on a k-lizard P is a member of the set {✓0, ✓1, ..., ✓2k1}, where ✓i = i⇡/k, and the angle between two sides is a measure of the interior angle of the k-lizard, and will also be a member of the set {✓0, ✓1, ..., ✓2k1}.

θ0 θ1 θ0 θ1 θ2 θ1 θ0 θ2 θ3 θ2 θ0 θ1 θ3 θ4

A reflex angle is an angle of measure larger than ⇡ but less than 2⇡.

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Definitions The direction of a side on a k-lizard P is a member of the set {✓0, ✓1, ..., ✓2k1}, where ✓i = i⇡/k, and the angle between two sides is a measure of the interior angle of the k-lizard, and will also be a member of the set {✓0, ✓1, ..., ✓2k1}.

θ0 θ1 θ0 θ1 θ2 θ1 θ0 θ2 θ3 θ2 θ0 θ1 θ3 θ4

A reflex angle is an angle of measure larger than ⇡ but less than 2⇡.

θ1 Willamette REU k-MSP Intersection Graphs August 4, 2017 7 / 67

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Definitions The direction of a side on a k-lizard P is a member of the set {✓0, ✓1, ..., ✓2k1}, where ✓i = i⇡/k, and the angle between two sides is a measure of the interior angle of the k-lizard, and will also be a member of the set {✓0, ✓1, ..., ✓2k1}.

θ0 θ1 θ0 θ1 θ2 θ1 θ0 θ2 θ3 θ2 θ0 θ1 θ3 θ4

A reflex angle is an angle of measure larger than ⇡ but less than 2⇡.

θ1 θ2 Willamette REU k-MSP Intersection Graphs August 4, 2017 7 / 67

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Definitions The direction of a side on a k-lizard P is a member of the set {✓0, ✓1, ..., ✓2k1}, where ✓i = i⇡/k, and the angle between two sides is a measure of the interior angle of the k-lizard, and will also be a member of the set {✓0, ✓1, ..., ✓2k1}.

θ0 θ1 θ0 θ1 θ2 θ1 θ0 θ2 θ3 θ2 θ0 θ1 θ3 θ4

A reflex angle is an angle of measure larger than ⇡ but less than 2⇡.

θ1 θ5 θ2 Willamette REU k-MSP Intersection Graphs August 4, 2017 7 / 67

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Definitions A region Q ✓ P is a maximal convex sub-polygon, or scale, of P if Q is a convex k-lizard that is maximal in P.

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Definitions A region Q ✓ P is a maximal convex sub-polygon, or scale, of P if Q is a convex k-lizard that is maximal in P.

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Definitions A region Q ✓ P is a maximal convex sub-polygon, or scale, of P if Q is a convex k-lizard that is maximal in P.

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Definitions A region Q ✓ P is a maximal convex sub-polygon, or scale, of P if Q is a convex k-lizard that is maximal in P. A graph G is a k-maximal sub-polygon graph or k-MSP graph if it is the intersection graph of the scales of a k-lizard.

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Convex Subset Lemma Lemma Let R be a convex k-lizard contained within a k-lizard L. Then R is contained within at least one scale of L.

R S

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Convex Subset Lemma Lemma Let R be a convex k-lizard contained within a k-lizard L. Then R is contained within at least one scale of L.

R S

Proof.

If R is maximal, then R is a scale. If R is not maximal, then we can extend the sides of R in at least one of the k directions to a side of L to create S within which R is contained.

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A reflex angle in 3-MSP Consider a ✓4 angle in the following 3-lizard. There are 3 scales touching this reflex angle.

θ4 Willamette REU k-MSP Intersection Graphs August 4, 2017 10 / 67

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A reflex angle in 3-MSP Consider a ✓4 angle in the following 3-lizard. There are 3 scales touching this reflex angle.

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A reflex angle in 3-MSP Consider a ✓4 angle in the following 3-lizard. There are 3 scales touching this reflex angle.

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A reflex angle in 3-MSP Consider a ✓4 angle in the following 3-lizard. There are 3 scales touching this reflex angle.

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A reflex angle in 3-MSP Consider a ✓4 angle in the following 3-lizard. There are 3 scales touching this reflex angle.

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A reflex angle in 3-MSP Consider a ✓4 angle in the following 3-lizard. There are 3 scales touching this reflex angle. We define a proto-scale to be a line segment contained in the interior

  • f the k-lizard in a ✓i direction which touches the boundary of the

k-lizard at a reflex angle.

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Definitions Let r be a reflex angle in a k-lizard L with internal angle measure ✓k+j. Then there exist j + 1 proto-scales, with each one contained in at least

  • ne scale.

θ7 A ✓k+3 reflex angle in 4-MSP yielding 4 proto-scales

Willamette REU k-MSP Intersection Graphs August 4, 2017 11 / 67

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Definitions Let r be a reflex angle in a k-lizard L with internal angle measure ✓k+j. Then there exist j + 1 proto-scales, with each one contained in at least

  • ne scale.

θ7 A ✓k+3 reflex angle in 4-MSP yielding 4 proto-scales

Willamette REU k-MSP Intersection Graphs August 4, 2017 11 / 67

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Definitions Let r be a reflex angle in a k-lizard L with internal angle measure ✓k+j. Then there exist j + 1 proto-scales, with each one contained in at least

  • ne scale.

θ7 A ✓k+3 reflex angle in 4-MSP yielding 4 proto-scales

Willamette REU k-MSP Intersection Graphs August 4, 2017 11 / 67

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Definitions Let r be a reflex angle in a k-lizard L with internal angle measure ✓k+j. Then there exist j + 1 proto-scales, with each one contained in at least

  • ne scale.

θ7 A ✓k+3 reflex angle in 4-MSP yielding 4 proto-scales

Willamette REU k-MSP Intersection Graphs August 4, 2017 11 / 67

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Definitions Let r be a reflex angle in a k-lizard L with internal angle measure ✓k+j. Then there exist j + 1 proto-scales, with each one contained in at least

  • ne scale.

p

θ7 A ✓k+3 reflex angle in 4-MSP yielding 4 proto-scales

Willamette REU k-MSP Intersection Graphs August 4, 2017 11 / 67

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Proto-scales Lemma We define a g-region of a k-lizard L as a region where g scales intersect. Lemma (Proto-scales) In a k-lizard at a reflex interior angle ✓k+j, there is a g-region, where g j + 1.

p

θ7

Willamette REU k-MSP Intersection Graphs August 4, 2017 12 / 67

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Proto-scales Lemma We define a g-region of a k-lizard L as a region where g scales intersect. Lemma (Proto-scales) In a k-lizard at a reflex interior angle ✓k+j, there is a g-region, where g j + 1.

p

θ7

Proof. By definition, at a reflex angle ✓k+j, there are j + 1 distinct proto-scales that each belong to at least one scale in L. Each of these proto-scales intersect at p, so the scales formed by them must intersect as well, so there is a g-region at p.

Willamette REU k-MSP Intersection Graphs August 4, 2017 12 / 67

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Ray Lemma Lemma Given three vertices a, b, c in a k-MSP graph with corresponding scales A, B, C such that a $ b and b $ c, but a = c, and from every point on the boundary of B A there exists a ray in a ✓i direction that intersects A, then there exists a scale D intersecting A, B, and C.

A B p r C

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Ray Lemma Lemma Given three vertices a, b, c in a k-MSP graph with corresponding scales A, B, C such that a $ b and b $ c, but a = c, and from every point on the boundary of B A there exists a ray in a ✓i direction that intersects A, then there exists a scale D intersecting A, B, and C.

A B p r C

Proof. Since b $ c, but a = c, C must intersect boundary(B A) at two distinct points. Consider a point p on the boundary of both B A and and C \ B and a ray r originating at p with ✓i direction that intersects A.

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Ray Lemma

A B C p l q p’ ε D

Let ✏ > 0 such that B✏(p) is contained within P. Since B is convex, we can extend r by length ✏ into C B at direction ✓k+i to form a line segment ` with one endpoint p0. The opposite endpoint of ` should lie at a point q in A By Convex Subset Lemma, ` is contained in at least one scale and since it cannot be wholly contained in A, B, or C, there must exist an additional scale D that contains `.

Willamette REU k-MSP Intersection Graphs August 4, 2017 14 / 67

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Parallel Sides Lemma Lemma Given scales A, B, C in a lizard, if a $ b, b $ c, and a and b have no shared neighbors, then the region B A C that borders both A and C has parallel boundary components s1 and s2 extending from the corners of A \ B to the corners of B \ C. A B C s1 s2

Willamette REU k-MSP Intersection Graphs August 4, 2017 15 / 67

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Parallel Sides Lemma Let 1 be a boundary component between B A and B \ A, and consider the angle ↵ formed where 1 reaches a side s2 of B. If ↵  ✓k2, then we can use the ray lemma to show that some scale intersects A and B. A B C s1 s2 δ1

A B s1 s2 δ1

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End Regions In a k-lizard L, we say that for a scales A and B, A \ B is an end region of A if A B is connected.

A B

A B

In a 3-lizard: an end region (left) and not an end region (right).

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End Regions In a k-lizard L, we say that for a scales A and B, A \ B is an end region of A if A B is connected.

A B

A B

In a 3-lizard: an end region (left) and not an end region (right).

On the left, A \ B is an end region of A and B. Additionally, A \ B is a 2-region since it is contained in 2 different scales. We then say A \ B is an end 2-region of A.

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End 2-Regions Lemma Lemma (End 2-Regions) Any scale has at most two end 2-regions.

A B C

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End 2-Regions Lemma Lemma (End 2-Regions) Any scale has at most two end 2-regions.

A B C

Proof. Suppose B has three end 2-regions; B \ A, B \ C, and B \ D.

Willamette REU k-MSP Intersection Graphs August 4, 2017 18 / 67

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End 2-Regions Lemma Lemma (End 2-Regions) Any scale has at most two end 2-regions.

A B C

Proof. Suppose B has three end 2-regions; B \ A, B \ C, and B \ D. The sides between B \ A and B \ C must be parallel by the Parallel Sides Lemma.

Willamette REU k-MSP Intersection Graphs August 4, 2017 18 / 67

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End 2-Regions Lemma Lemma (End 2-Regions) Any scale has at most two end 2-regions.

A B C D

Proof. Suppose B has three end 2-regions; B \ A, B \ C, and B \ D. The sides between B \ A and B \ C must be parallel by the Parallel Sides Lemma. The sides between B \ A and B \ D must be parallel; thus B \ D is either not an end region, or not a 2-region.

Willamette REU k-MSP Intersection Graphs August 4, 2017 18 / 67

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End 2-Regions Lemma Lemma (End 2-Regions) Any scale has at most two end 2-regions.

A B C D

Proof. Suppose B has three end 2-regions; B \ A, B \ C, and B \ D. The sides between B \ A and B \ C must be parallel by the Parallel Sides Lemma. The sides between B \ A and B \ D must be parallel; thus B \ D is either not an end region, or not a 2-region.

Willamette REU k-MSP Intersection Graphs August 4, 2017 18 / 67

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Summary of Lemmas Convex Subset Lemma Let R be a convex k-lizard contained within a k-lizard L. Then R is contained within at least one scale of L.

R S

Willamette REU k-MSP Intersection Graphs August 4, 2017 19 / 67

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Summary of Lemmas Convex Subset Lemma Let R be a convex k-lizard contained within a k-lizard L. Then R is contained within at least one scale of L. Proto-Scale Lemma In a k-lizard at a reflex interior angle ✓k+j, there is a g-region, where g j + 1.

p

θ7

Willamette REU k-MSP Intersection Graphs August 4, 2017 19 / 67

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Summary of Lemmas Ray Lemma Given scales A, B, C in a k-lizard, if a $ b, b $ c, and a = c, and from every point on the boundary of B A there exists a ray in a ✓i direction that intersects A, then there exists a scale D intersecting A, B, and C.

A B p r C

Willamette REU k-MSP Intersection Graphs August 4, 2017 20 / 67

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Summary of Lemmas Ray Lemma Given scales A, B, C in a k-lizard, if a $ b, b $ c, and a = c, and from every point on the boundary of B A there exists a ray in a ✓i direction that intersects A, then there exists a scale D intersecting A, B, and C. Parallel Sides Lemma Given scales A, B, C in a k-lizard, if a $ b, b $ c, and a and b have no shared neighbors, then the region B A C has parallel boundary components s1 and s2 from A \ B to B \ C.

A B C s1 s2

Willamette REU k-MSP Intersection Graphs August 4, 2017 20 / 67

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Summary of Lemmas Ray Lemma Given scales A, B, C in a k-lizard, if a $ b, b $ c, and a = c, and from every point on the boundary of B A there exists a ray in a ✓i direction that intersects A, then there exists a scale D intersecting A, B, and C. Parallel Sides Lemma Given scales A, B, C in a k-lizard, if a $ b, b $ c, and a and b have no shared neighbors, then the region B A C has parallel boundary components s1 and s2 from A \ B to B \ C. End 2-Region Lemma Any scale has at most two end 2-regions.

A B C

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Separating Examples

2-MSP 3-MSP 4-MSP

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Graphs that are k-MSP Graphs Proposition (Complete Graphs). The complete graph Kn is a k-MSP graph for all n, k 2 N, where k 2. K4 in 2-MSP , K5 in 3-MSP , and 4-MSP

Willamette REU k-MSP Intersection Graphs August 4, 2017 22 / 67

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Graphs that are k-MSP Graphs Proposition (Complete Graphs). The complete graph Kn is a k-MSP graph for all n, k 2 N, where k 2. K4 in 2-MSP , K5 in 3-MSP , and 4-MSP

Willamette REU k-MSP Intersection Graphs August 4, 2017 22 / 67

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Graphs that are k-MSP Graphs Proposition (Complete Graphs). The complete graph Kn is a k-MSP graph for all n, k 2 N, where k 2. K4 in 2-MSP , K5 in 3-MSP , and 4-MSP

Willamette REU k-MSP Intersection Graphs August 4, 2017 22 / 67

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Graphs that are k-MSP Graphs Proposition (Complete Graphs). The complete graph Kn is a k-MSP graph for all n, k 2 N, where k 2. K4 in 2-MSP , K5 in 3-MSP , and 4-MSP

Willamette REU k-MSP Intersection Graphs August 4, 2017 22 / 67

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Graphs that are k-MSP Graphs Proposition (Complete Graphs). The complete graph Kn is a k-MSP graph for all n, k 2 N, where k 2. K4 in 2-MSP , K5 in 3-MSP , and 4-MSP

Willamette REU k-MSP Intersection Graphs August 4, 2017 22 / 67

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Graphs that are k-MSP Graphs Proposition (Complete Graphs). The complete graph Kn is a k-MSP graph for all n, k 2 N, where k 2. K4 in 2-MSP , K5 in 3-MSP , and 4-MSP

Willamette REU k-MSP Intersection Graphs August 4, 2017 22 / 67

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Graphs that are k-MSP Graphs Proposition (Complete Graphs). The complete graph Kn is a k-MSP graph for all n, k 2 N, where k 2. For k = 2, construct the "staircase" as shown previously, with n "stairs". For k 3, construct the first scale A as a triangle with two ✓1 angles and one ✓k2 angle. Add n 1 bumps to the longest side of A, alternating directions of the other two sides of A.

θ1 θk-2 θ1 θk-2

The construction of the 6-MSP representation of K4.

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Graphs that are k-MSP Graphs Proposition (Path Graphs). Pj is a k-MSP graph for all j, k 2 N, and k 2.

The 2- and 3-MSP representation of P6.

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Graphs that are k-MSP Graphs Proposition (Path Graphs). Pj is a k-MSP graph for all j, k 2 N, and k 2.

The 2- and 3-MSP representation of P6.

For all k, construct the k-lizard with consecutive parallelograms Q1, ..., Qn, intersecting such that for all 1  a  n, Qa [ Qa+1 is a polygon containing one reflex angle with measure ✓k+1.

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Separating Examples

2-MSP 3-MSP 4-MSP

Kn Pn

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Cycles are not k-MSP Graphs Theorem Cj is not a k-MSP graph when j 4 and k 2.

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Cycles are not k-MSP Graphs Theorem Cj is not a k-MSP graph when j 4 and k 2. Proof. Assume there exists a k-MSP representation of Cj. j scales must intersect two "neighbor" scales. Connect one point in each intersection with line segments. Note that each line segment must be contained in one of the j scales. The j points and j line segments must form a closed path P, which must be contained in the k-lizard.

P A B C

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Cycles are not k-MSP Graphs Theorem Cj is not a k-MSP graph when j 4 and k 2. Proof cont. Construct a convex k-lizard R such that R ✓ P is maximal inside P. The vertices of R must intersect at least three distinct sides of P, thus R must contain points from at least three distinct scales.

P R

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Cycles are not k-MSP Graphs Theorem Cj is not a k-MSP graph when j 4 and k 2. Proof cont. R must be contained in a scale S of L by the Convex Subset Lemma. S corresponds to a vertex with degree at least three, but all vertices in Cj must have degree two. So Cj is not a k-MSP graph.

R S

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Separating Examples

2-MSP 3-MSP 4-MSP

Kn Pn Cn (n>3)

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Induced Cycles in k-MSP Graphs Theorem C5 is an induced subgraph for 3 and 4-MSP but not for 2-MSP . Proof (C5 is not an induced subgraph for 2-MSP). Recall from Shearer 1982 and Maire 1993, all 2-MSP graphs are perfect. C5 is not perfect, so it cannot be a subgraph of a 2-MSP graph.

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Induced Cycles in k-MSP Graphs Theorem C5 is an induced subgraph for 3 and 4-MSP but not for 2-MSP . Proof (C5 is not an induced subgraph for 2-MSP). Recall from Shearer 1982 and Maire 1993, all 2-MSP graphs are perfect. C5 is not perfect, so it cannot be a subgraph of a 2-MSP graph. Maire also proved that C4 is the largest induced cycle that can be made in 2-MSP .

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Induced Cycles in k-MSP Graphs Theorem C5 is an induced subgraph for 3 and 4-MSP but not for 2-MSP . Proof (C5 is an induced subgraph for 3 and 4-MSP). See the constructions for an induced C5 with 3 and 4-MSP .

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Separating Examples

2-MSP 3-MSP 4-MSP

Kn Pn Cn (n>3) induced C5

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Induced Cycles in k-MSP Graphs Proposition Cn is an induced subgraph of a k-MSP graph when n 3 and k 5. Along one "direction," alternate two types of triangles: one with angle measures (✓1, ✓1, ✓k2), and another with angle measures (✓1, ✓2, ✓k3) Construct n 3 of these triangles, then complete the cycle with 3 paralellograms

θ1 θk-3 θ2 θ1 θ1 θk-2

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SLIDE 76

Induced Cycles in k-MSP Graphs

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SLIDE 77

Induced Cycles in k-MSP Graphs Conjecture In 3-MSP and 4-MSP , there is an upper bound on the size of the largest possible induced cycle. Alternating triangle construction fails because there is only one triangle possible in each case (60, 60, 60 in 3-MSP or 45, 45, 90 in 4-MSP) k-MSP Maximum Induced Cycle 2-MSP 4 (Maire) 3-MSP 12 4-MSP 16 (5+)-MSP unbounded

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SLIDE 78

Tree Graphs Theorem All tree graphs are 2-MSP graphs.

a b c d e A B C D E

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SLIDE 79

Tree Graphs Theorem All tree graphs are 2-MSP graphs.

a b c d e f A B C D E F

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SLIDE 80

Tree Graphs Theorem All tree graphs are 2-MSP graphs.

a b c d e f g A B C D E F G

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SLIDE 81

Tree Graphs Theorem Tree graphs are k-MSP graphs for all k 3 if and only if they are caterpillars.

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SLIDE 82

Tree Graphs Theorem Tree graphs are k-MSP graphs for all k 3 if and only if they are caterpillars. Proof (()

Recall the construction for Pn for any k. We can extend any scale corresponding to any vertex of P. On any of the scales, we can add triangles with interior angles equal to ✓k+1.

A 3-lizard whose graph is a caterpillar

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SLIDE 83

Tree Graphs Proof ()), by contradiction. a b c d e f g Recall that NC7 (the smallest non-caterpillar with 7 vertices) is a subgraph of every tree that is not a caterpillar, so it suffices to show that NC7 is not a k-MSP graph for all k 3. By the End 2-Regions Lemma, the corresponding scale C has at most two end 2-regions and has parallel sides.

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SLIDE 84

Tree Graphs Proof ()) cont. By the Proto-scales Lemma, the only angle that creates a 2-region is ✓k+1.

θk+1 θk+1

C F

Note that from any point on boundary(F C), there exists a ray in a ✓i direction that intersects C. By the Ray Lemma, we cannot form another scale intersecting F without creating a scale that intersects both C and F. So a k-MSP tree graph G must be a caterpillar.

Willamette REU k-MSP Intersection Graphs August 4, 2017 39 / 67

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SLIDE 85

Separating Examples

2-MSP 3-MSP 4-MSP

Kn Pn NC7 Cn (n>3) induced C5

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SLIDE 86

Seagull Graphs We define the family of seagull graphs, Seagulln, to be a clique with n vertices joined to the third vertex of P5, with a pendant vertex adjacent to a different vertex of the clique.

Seagull2(NC7) (left) and Seagull3 (right)

Willamette REU k-MSP Intersection Graphs August 4, 2017 41 / 67

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SLIDE 87

Seagull Graphs Theorem Seagullk1 is not a k-MSP graph, but is a (k 1)-MSP graph.

Willamette REU k-MSP Intersection Graphs August 4, 2017 42 / 67

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SLIDE 88

Seagull Graphs Theorem Seagullk1 is not a k-MSP graph, but is a (k 1)-MSP graph. Proof (Seagullk1 is not a k-MSP graph).

C G

{

Kk-1

Since C has two end 2-regions, it must have parallel sides between them (by Parallel Sides Lemma), so C G cannot be an end region. Therefore, we just need to consider the reflex angles at C [ G.

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SLIDE 89

Seagull Graphs Case I: C [ G has one reflex angle.

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SLIDE 90

Seagull Graphs Case I: C [ G has one reflex angle. G must intersect C at at least two points, one of which is the reflex angle. To avoid second reflex angle, G must intersect at a corner of C. However, this causes G to intersect an end 2-region of C, making it no longer a 2-region. θk+1

C D G

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SLIDE 91

Seagull Graphs Case II: C [ G has two reflex angles with measures ✓k+i and ✓k+j.

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SLIDE 92

Seagull Graphs Case II: C [ G has two reflex angles with measures ✓k+i and ✓k+j. We do not want to create any more than k 2 scales (not including C) for the k 1 clique of the seagull, so each angle cannot create more than k 2 proto-scales. This implies the maximum value for either i or j is k 3. The minimum value for i or j is 1, else the angles would not be reflex. We then split Case II into three subcases.

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SLIDE 93

Seagull Graphs Case II.a: i + j < k

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SLIDE 94

Seagull Graphs Case II.a: i + j < k

θk+i θk+j C G

If i + j < k, the sides of G are not parallel and point towards each

  • ther.

This means G C forms a shape such that from every point on the boundary of G C, we can construct a ray that intersects C. By the Ray Lemma, this means any scale intersecting G would also intersect C.

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SLIDE 95

Seagull Graphs Case II.b: i + j = k If i + j = k, the sides of G are parallel. We create ij scales with construction of G.

C

θk+2 θk+3

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SLIDE 96

Seagull Graphs Case II.b: i + j = k If i + j = k, the sides of G are parallel. We create ij scales with construction of G.

C

θk+2 θk+3

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SLIDE 97

Seagull Graphs Case II.b: i + j = k If i + j = k, the sides of G are parallel. We create ij scales with construction of G.

C

θk+2 θk+3

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SLIDE 98

Seagull Graphs Case II.b: i + j = k If i + j = k, the sides of G are parallel. We create ij scales with construction of G.

C

θk+2 θk+3

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SLIDE 99

Seagull Graphs Case II.b: i + j = k If i + j = k, the sides of G are parallel. We create ij scales with construction of G.

C

θk+2 θk+3

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SLIDE 100

Seagull Graphs Case II.b: i + j = k If i + j = k, the sides of G are parallel. We create ij scales with construction of G.

C

θk+2 θk+3

Willamette REU k-MSP Intersection Graphs August 4, 2017 46 / 67

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SLIDE 101

Seagull Graphs Case II.b: i + j = k If i + j = k, the sides of G are parallel. We create ij scales with construction of G.

C

θk+2 θk+3

Willamette REU k-MSP Intersection Graphs August 4, 2017 46 / 67

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SLIDE 102

Seagull Graphs Case II.b: i + j = k If i + j = k, the sides of G are parallel. We create ij scales with construction of G.

C

θk+2 θk+3

Willamette REU k-MSP Intersection Graphs August 4, 2017 46 / 67

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SLIDE 103

Seagull Graphs Recall that we cannot create any more than k 2 scales, so we need ij  k 2. Since i + j = k, without a loss of generality, we can make the substitution i = k j into the inequality to yield j k 2. However, we know that j must be smaller than k 3. )(

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SLIDE 104

Seagull Graphs Case II.c: i + j > k, in other words i + j k + 1

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SLIDE 105

Seagull Graphs Case II.c: i + j > k, in other words i + j k + 1

C G

θk+i

θk+j

If i + j k + 1, the sides of G are not parallel and point in directions away from each other. WLOG, the minimum number for i is 4, since if i were any smaller, i + j 6 k + 1.

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SLIDE 106

Seagull Graphs Case II.c: i + j > k, in other words i + j k + 1

C G

θk+i

θk+j

If i + j k + 1, the sides of G are not parallel and point in directions away from each other. WLOG, the minimum number for i is 4, since if i were any smaller, i + j 6 k + 1. If i = 4, then 4 + j k + 1, which implies j k 3. However, we know j  k 3, so we just consider the case where j = k 3. After substitution, we determine that ij > k 1. )(

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SLIDE 107

Seagull Graphs Case III: There are three or more reflex angles at C [ G.

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SLIDE 108

Seagull Graphs Case III: There are three or more reflex angles at C [ G. We know that no matter the angle measurements, any two reflex angles create too many scales. Adding an additional reflex angle will never remove scales. Thus, it directly follows that three or more reflex angles would also create too many scales. Therefore, Seagullk1 is not a k-MSP graph.

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SLIDE 109

Seagull Graphs Proof (Seagullk1 is a (k 1)-MSP graph.) We propose the following construction: Let m = k 1. Construct the representation for P5 in m-MSP .

Willamette REU k-MSP Intersection Graphs August 4, 2017 50 / 67

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SLIDE 110

Seagull Graphs Construct G such that the two reflex angles at the intersection are ✓2m1 and ✓m+1.

θ2m-1 θm+1

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SLIDE 111

Seagull Graphs Construct G such that the two reflex angles at the intersection are ✓2m1 and ✓m+1.

θ2m-1 θm+1

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SLIDE 112

Seagull Graphs Construct G such that the two reflex angles at the intersection are ✓2m1 and ✓m+1.

θ2m-1 θm+1

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SLIDE 113

Seagull Graphs Construct G such that the two reflex angles at the intersection are ✓2m1 and ✓m+1.

θ2m-1 θm+1

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SLIDE 114

Seagull Graphs Construct G such that the two reflex angles at the intersection are ✓2m1 and ✓m+1.

θ2m-1 θm+1

We have formed m = k 1 total scales.

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SLIDE 115

Seagull Graphs We know that we can create the pendant vertex since G has parallel sides.

θ2m-1 θm+1

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SLIDE 116

Seagull Graphs Example. Seagull3 is a 2- and 3-MSP graph but not a 4-MSP graph.

a b c d e f g h A B C D E F G H

A B C D E F G H

Conjecture Seagullk is a j-MSP graph for all j  k.

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SLIDE 117

Separating Examples

2-MSP 3-MSP 4-MSP

Kn Pn NC7 Seagull3 Cn (n>3) induced C5

Willamette REU k-MSP Intersection Graphs August 4, 2017 54 / 67

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SLIDE 118

Turtle Graphs We define the family of turtle graphs, Turtlen, p, to be a clique with n vertices, to which p pendant vertices are attached, each to distinct vertices of the clique.

Turtle6,4

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SLIDE 119

Turtle Graphs Theorem Turtlen,k is a k-MSP graph for all k and some n.

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SLIDE 120

Turtle Graphs Theorem Turtlen,k is a k-MSP graph for all k and some n.

Turtle9,4 in 4-MSP

Willamette REU k-MSP Intersection Graphs August 4, 2017 56 / 67

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SLIDE 121

Turtle Graphs Theorem Turtlen,k is a k-MSP graph for all k and some n.

Turtle9,4 in 4-MSP

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SLIDE 122

Turtle Graphs Theorem Turtlen,k is a k-MSP graph for all k and some n.

Turtle9,4 in 4-MSP

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SLIDE 123

Turtle Graphs Theorem Turtlen,k is a k-MSP graph for all k and some n.

Turtle9,4 in 4-MSP

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SLIDE 124

Turtle Graphs Theorem Turtlen,k+1 is not a k-MSP graph for any n 2 N Proof. Assume by way of contradiction that there exists a k-lizard which gives the k-MSP graph Turtlen,k+1. Let p1, ..., pk+1 be the pendant vertices in Turtlen,k+1, and q1, ..., qk+1 the vertices in the graph adjacent to each pendant vertex.

pa qa qb pb

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SLIDE 125

Turtle Graphs Theorem Turtlen,k+1 is not a k-MSP graph for any n 2 N Proof cont. By the parallel sides lemma, since pa $ qa, and qa has another neighbor qb, Qa Pa Qb contains a pair of parallel sides. For each qa, 1  a  k + 1, the lemma holds, so we have k + 1 pairs

  • f parallel sides. Since we have k

directions for the sides, two pairs of parallel sides share a direction.

pa qa qb pb s1 s2 s3 s4 Qa Qb

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SLIDE 126

Turtle Graphs Theorem Turtlen,k+1 is not a k-MSP graph for any n 2 N Proof cont. Case 1: If the sides are arranged s1, s2, s3, s4, then Qa and Qb do not intersect.

pa qa qb pb s1 s2 s3 s4 Qa Qb

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SLIDE 127

Turtle Graphs Theorem Turtlen,k+1 is not a k-MSP graph for any n 2 N Proof cont. Case 1: If the sides are arranged s1, s2, s3, s4, then Qa and Qb do not intersect. Case 2: If the sides are arranged s1, s3, s2, s4, then there is a scale Qc between s3 and s2. By the ray lemma, a scale attached to Qa also intersects Qc, so Qa cannot have a pendant vertex.

pa qa qb pb

s1 s2 s3 s4 Qa Qb Pb Pa Qc

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SLIDE 128

Turtle Graphs Theorem Turtlen,k+1 is not a k-MSP graph for any n 2 N Proof cont Case 3: If the sides are arranged s1, s3, s4, s2, then we can draw a ray from any point along s1 or s2 that intersects Qa and Qb. By the ray lemma, a scale Pa intersecting Qa creates a scale that intersects Pa, Qa, and Qc, so Qa cannot have a pendant vertex.

pa qa qb pb

s1 s2 s3 s4 Qb Qa Pa

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SLIDE 129

Turtle Graphs Theorem Turtlen,k+1 is not a k-MSP graph for any n 2 N Proof cont Case 4: If two sides intersect, we have two subcases:

If in s1, s2, s3, s4, if s2 and s3 intersect, then Qa and Qb don’t intersect. If in s1, s3, s4, s2, if s1 and s3 intersect, then again by the ray lemma, a scale Pa intersecting Qa creates a scale that intersects Pa, Qa, and Qb, so Qa cannot have a pendant vertex.

s1 s2 s3 s4 Qa Qb

s1 s2 s3 s4 Qb Qa Pa

Willamette REU k-MSP Intersection Graphs August 4, 2017 61 / 67

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SLIDE 130

Separating Examples

2-MSP 3-MSP 4-MSP

Kn Pn NC7 Seagull3 Cn (n>3) Turtlen,4

Turtlen,3 induced C5

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SLIDE 131

Separating Examples

2-MSP 3-MSP 4-MSP

Kn Pn NC7 Seagull3 Cn (n>3) ??? ??? Turtlen,4

Turtlen,3 induced C5

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SLIDE 132

Generalized Separating Examples

(k-1)-MSP k-MSP (k+1)-MSP

Kn Pn Seagullk-1 Seagullk Cn (n>3) Turtlen, k Ø Turtle Seagulln, k Turtlen, k+1

Willamette REU k-MSP Intersection Graphs August 4, 2017 64 / 67

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SLIDE 133

Seagull-Turtle Graph Conjecture The Seagull-Turtle Graph, SeaTurtk,n,k, is a k-MSP graph but not j-MSP for j 6= k.

Willamette REU k-MSP Intersection Graphs August 4, 2017 65 / 67

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SLIDE 134

Seagull-Turtle Graph Conjecture The Seagull-Turtle Graph, SeaTurtk,n,k, is a k-MSP graph but not j-MSP for j 6= k.

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SLIDE 135

Future Research There are several other things we would like to consider: Considering k-snakes, which require integer side lengths (Kaplan, 2017) Requiring maximal sub-polygons also have integer side lengths

Does 4-MSP reduce to 2-MSP?

Non-simply connected k-lizards

Willamette REU k-MSP Intersection Graphs August 4, 2017 66 / 67

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SLIDE 136

Acknowledgements We gratefully acknowledge: Josh Laison for his invaluable contributions to the project Richard Moy for his ideas, uplifting spirit and encouragement to have fun Erin McNicholas and Josh Laison for organizing the Willamette Mathematics Consortium REU The support of NSF grant DMS 1460982

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