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

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 of the k -lizard in a ✓ i direction which touches the boundary of the k -lizard at a reflex angle. Willamette REU k -MSP Intersection Graphs August 4, 2017 10 / 67

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 one scale. θ 7 A ✓ k + 3 reflex angle in 4-MSP yielding 4 proto-scales Willamette REU k -MSP Intersection Graphs August 4, 2017 11 / 67

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 one scale. θ 7 A ✓ k + 3 reflex angle in 4-MSP yielding 4 proto-scales Willamette REU k -MSP Intersection Graphs August 4, 2017 11 / 67

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 one scale. θ 7 A ✓ k + 3 reflex angle in 4-MSP yielding 4 proto-scales Willamette REU k -MSP Intersection Graphs August 4, 2017 11 / 67

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 one scale. θ 7 A ✓ k + 3 reflex angle in 4-MSP yielding 4 proto-scales Willamette REU k -MSP Intersection Graphs August 4, 2017 11 / 67

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 one scale. p θ 7 A ✓ k + 3 reflex angle in 4-MSP yielding 4 proto-scales Willamette REU k -MSP Intersection Graphs August 4, 2017 11 / 67

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 θ 7 Willamette REU k -MSP Intersection Graphs August 4, 2017 12 / 67

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 θ 7 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

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. C B p r A Willamette REU k -MSP Intersection Graphs August 4, 2017 13 / 67

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. C B p r A 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 . Willamette REU k -MSP Intersection Graphs August 4, 2017 13 / 67

Ray Lemma C B D p’ ε p l A q 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 p 0 . 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

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 s 1 and s 2 extending from the corners of A \ B to the corners of B \ C. A s 1 B s 2 C Willamette REU k -MSP Intersection Graphs August 4, 2017 15 / 67

Parallel Sides Lemma A s 1 Let � 1 be a boundary δ 1 B component between B � A s 2 and B \ A , and consider the C angle ↵ formed where � 1 reaches a side s 2 of B . If ↵  ✓ k � 2 , then we can use A s 1 the ray lemma to show that some scale intersects A and δ 1 B B . s 2 Willamette REU k -MSP Intersection Graphs August 4, 2017 16 / 67

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. B B A A In a 3-lizard: an end region (left) and not an end region (right). Willamette REU k -MSP Intersection Graphs August 4, 2017 17 / 67

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. B B A A 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 . Willamette REU k -MSP Intersection Graphs August 4, 2017 17 / 67

End 2-Regions Lemma Lemma (End 2-Regions) Any scale has at most two end 2-regions. A C B Willamette REU k -MSP Intersection Graphs August 4, 2017 18 / 67

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

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

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

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

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

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 θ 7 Willamette REU k -MSP Intersection Graphs August 4, 2017 19 / 67

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 . C B p r A Willamette REU k -MSP Intersection Graphs August 4, 2017 20 / 67

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 s 1 and s 2 from A \ B to B \ C . A s 1 B s 2 C Willamette REU k -MSP Intersection Graphs August 4, 2017 20 / 67

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 s 1 and s 2 from A \ B to B \ C . End 2-Region Lemma Any scale has at most two end 2-regions. A C B Willamette REU k -MSP Intersection Graphs August 4, 2017 20 / 67

Separating Examples 2-MSP 3-MSP 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 21 / 67

Graphs that are k -MSP Graphs Proposition (Complete Graphs). The complete graph K n is a k -MSP graph for all n , k 2 N , where k � 2. K 4 in 2-MSP , K 5 in 3-MSP , and 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 22 / 67

Graphs that are k -MSP Graphs Proposition (Complete Graphs). The complete graph K n is a k -MSP graph for all n , k 2 N , where k � 2. K 4 in 2-MSP , K 5 in 3-MSP , and 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 22 / 67

Graphs that are k -MSP Graphs Proposition (Complete Graphs). The complete graph K n is a k -MSP graph for all n , k 2 N , where k � 2. K 4 in 2-MSP , K 5 in 3-MSP , and 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 22 / 67

Graphs that are k -MSP Graphs Proposition (Complete Graphs). The complete graph K n is a k -MSP graph for all n , k 2 N , where k � 2. K 4 in 2-MSP , K 5 in 3-MSP , and 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 22 / 67

Graphs that are k -MSP Graphs Proposition (Complete Graphs). The complete graph K n is a k -MSP graph for all n , k 2 N , where k � 2. K 4 in 2-MSP , K 5 in 3-MSP , and 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 22 / 67

Graphs that are k -MSP Graphs Proposition (Complete Graphs). The complete graph K n is a k -MSP graph for all n , k 2 N , where k � 2. K 4 in 2-MSP , K 5 in 3-MSP , and 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 22 / 67

Graphs that are k -MSP Graphs Proposition (Complete Graphs). The complete graph K n 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 ✓ k � 2 angle. Add n � 1 bumps to the longest side of A , alternating directions of the other two sides of A . θ 1 θ 1 θ k-2 θ k-2 The construction of the 6-MSP representation of K 4 . Willamette REU k -MSP Intersection Graphs August 4, 2017 23 / 67

Graphs that are k -MSP Graphs Proposition (Path Graphs). P j is a k -MSP graph for all j , k 2 N , and k � 2. The 2- and 3-MSP representation of P 6 . Willamette REU k -MSP Intersection Graphs August 4, 2017 24 / 67

Graphs that are k -MSP Graphs Proposition (Path Graphs). P j is a k -MSP graph for all j , k 2 N , and k � 2. The 2- and 3-MSP representation of P 6 . For all k , construct the k -lizard with consecutive parallelograms Q 1 , ..., Q n , intersecting such that for all 1  a  n , Q a [ Q a + 1 is a polygon containing one reflex angle with measure ✓ k + 1 . Willamette REU k -MSP Intersection Graphs August 4, 2017 24 / 67

Separating Examples 2-MSP 3-MSP K n P n 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 25 / 67

Cycles are not k -MSP Graphs Theorem C j is not a k-MSP graph when j � 4 and k � 2 . Willamette REU k -MSP Intersection Graphs August 4, 2017 26 / 67

Cycles are not k -MSP Graphs Theorem C j is not a k-MSP graph when j � 4 and k � 2 . Proof. Assume there exists a k -MSP representation of C j . j scales must intersect two "neighbor" scales. Connect one point in each intersection with line segments. P A C Note that each line segment must be contained in one of the j scales. B The j points and j line segments must form a closed path P , which must be contained in the k -lizard. Willamette REU k -MSP Intersection Graphs August 4, 2017 26 / 67

Cycles are not k -MSP Graphs Theorem C j 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 P least three distinct sides of P , thus R R must contain points from at least three distinct scales. Willamette REU k -MSP Intersection Graphs August 4, 2017 27 / 67

Cycles are not k -MSP Graphs Theorem C j is not a k-MSP graph when j � 4 and k � 2 . Proof cont. R must be contained in a scale S of S L by the Convex Subset Lemma. S corresponds to a vertex with degree at least three, but all vertices R in C j must have degree two. So C j is not a k -MSP graph. Willamette REU k -MSP Intersection Graphs August 4, 2017 28 / 67

Separating Examples 2-MSP 3-MSP K n P n C n (n>3) 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 29 / 67

Induced Cycles in k -MSP Graphs Theorem C 5 is an induced subgraph for 3 and 4-MSP but not for 2-MSP . Proof ( C 5 is not an induced subgraph for 2-MSP). Recall from Shearer 1982 and Maire 1993, all 2-MSP graphs are perfect. C 5 is not perfect, so it cannot be a subgraph of a 2-MSP graph. Willamette REU k -MSP Intersection Graphs August 4, 2017 30 / 67

Induced Cycles in k -MSP Graphs Theorem C 5 is an induced subgraph for 3 and 4-MSP but not for 2-MSP . Proof ( C 5 is not an induced subgraph for 2-MSP). Recall from Shearer 1982 and Maire 1993, all 2-MSP graphs are perfect. C 5 is not perfect, so it cannot be a subgraph of a 2-MSP graph. Maire also proved that C 4 is the largest induced cycle that can be made in 2-MSP . Willamette REU k -MSP Intersection Graphs August 4, 2017 30 / 67

Induced Cycles in k -MSP Graphs Theorem C 5 is an induced subgraph for 3 and 4-MSP but not for 2-MSP . Proof ( C 5 is an induced subgraph for 3 and 4-MSP). See the constructions for an induced C 5 with 3 and 4-MSP . Willamette REU k -MSP Intersection Graphs August 4, 2017 31 / 67

Separating Examples 2-MSP 3-MSP K n P n induced C 5 C n (n>3) 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 32 / 67

Induced Cycles in k -MSP Graphs Proposition C n 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 , ✓ k � 2 ) , and another with angle measures ( ✓ 1 , ✓ 2 , ✓ k � 3 ) Construct n � 3 of these triangles, then complete the cycle with 3 paralellograms θ 1 θ k-3 θ k-2 θ 2 θ 1 θ 1 Willamette REU k -MSP Intersection Graphs August 4, 2017 33 / 67

Induced Cycles in k -MSP Graphs Willamette REU k -MSP Intersection Graphs August 4, 2017 34 / 67

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 Willamette REU k -MSP Intersection Graphs August 4, 2017 35 / 67

Tree Graphs Theorem All tree graphs are 2-MSP graphs. a b c D E d e B C A Willamette REU k -MSP Intersection Graphs August 4, 2017 36 / 67

Tree Graphs Theorem All tree graphs are 2-MSP graphs. a b c D E C d e f B F A Willamette REU k -MSP Intersection Graphs August 4, 2017 36 / 67

Tree Graphs Theorem All tree graphs are 2-MSP graphs. a G b c D E C d e f B F g A Willamette REU k -MSP Intersection Graphs August 4, 2017 36 / 67

Tree Graphs Theorem Tree graphs are k-MSP graphs for all k � 3 if and only if they are caterpillars. Willamette REU k -MSP Intersection Graphs August 4, 2017 37 / 67

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 P n 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 Willamette REU k -MSP Intersection Graphs August 4, 2017 37 / 67

Tree Graphs Proof ( ) ), by contradiction. g f a b c d e Recall that NC 7 (the smallest non-caterpillar with 7 vertices) is a subgraph of every tree that is not a caterpillar, so it suffices to show that NC 7 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. Willamette REU k -MSP Intersection Graphs August 4, 2017 38 / 67

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

Separating Examples 2-MSP 3-MSP NC 7 K n P n induced C 5 C n (n>3) 4-MSP Willamette REU k -MSP Intersection Graphs August 4, 2017 40 / 67

Seagull Graphs We define the family of seagull graphs , Seagull n , to be a clique with n vertices joined to the third vertex of P 5 , with a pendant vertex adjacent to a different vertex of the clique. Seagull 2 ( NC 7 ) (left) and Seagull 3 (right) Willamette REU k -MSP Intersection Graphs August 4, 2017 41 / 67

Seagull Graphs Theorem Seagull k � 1 is not a k-MSP graph, but is a ( k � 1 ) -MSP graph. Willamette REU k -MSP Intersection Graphs August 4, 2017 42 / 67

Seagull Graphs Theorem Seagull k � 1 is not a k-MSP graph, but is a ( k � 1 ) -MSP graph. Proof ( Seagull k � 1 is not a k -MSP graph). K k -1 { G C 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 . Willamette REU k -MSP Intersection Graphs August 4, 2017 42 / 67

Seagull Graphs Case I: C [ G has one reflex angle. Willamette REU k -MSP Intersection Graphs August 4, 2017 43 / 67

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. D G θ k+1 C Willamette REU k -MSP Intersection Graphs August 4, 2017 43 / 67

Seagull Graphs Case II: C [ G has two reflex angles with measures ✓ k + i and ✓ k + j . Willamette REU k -MSP Intersection Graphs August 4, 2017 44 / 67

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. Willamette REU k -MSP Intersection Graphs August 4, 2017 44 / 67

Seagull Graphs Case II.a: i + j < k Willamette REU k -MSP Intersection Graphs August 4, 2017 45 / 67

Seagull Graphs Case II.a: i + j < k G θ k+i θ k+j C If i + j < k , the sides of G are not parallel and point towards each other. 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 . Willamette REU k -MSP Intersection Graphs August 4, 2017 45 / 67

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 . θ k+ 3 θ k+ 2 C Willamette REU k -MSP Intersection Graphs August 4, 2017 46 / 67

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 θ k+ 2 θ k+ 3 Willamette REU k -MSP Intersection Graphs August 4, 2017 46 / 67

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 θ k+ 2 θ k+ 3 Willamette REU k -MSP Intersection Graphs August 4, 2017 46 / 67

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 θ k+ 2 θ k+ 3 Willamette REU k -MSP Intersection Graphs August 4, 2017 46 / 67

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 θ k+ 2 θ k+ 3 Willamette REU k -MSP Intersection Graphs August 4, 2017 46 / 67

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 θ k+ 2 θ k+ 3 Willamette REU k -MSP Intersection Graphs August 4, 2017 46 / 67