there and back again, a t-segment...

There and Back Again, A T-Segment Tale

Fugatzs, Hokets, Dongles & Shenanigans

Melinda Kleinman

Willamette University, REU-RET

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 1 / 14

Graph Theory

The study of graphs, in this context, a collection of vertices and acollection of edges that connect pairs of vertices.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 2 / 14

Graph Theory

The study of graphs, in this context, a collection of vertices and acollection of edges that connect pairs of vertices.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 2 / 14

T-Segment GraphA graph that can be represented as follows :

Each vertex in the graph is represented with a line segment,Each edge between two vertices is represented by an intersectionof their segments.BUT the point of intersection is not an interior point of both linesegments.

Thus creating a "T" intersection.

ba

a

b

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 3 / 14

T-Segment GraphA graph that can be represented as follows :

Each vertex in the graph is represented with a line segment,Each edge between two vertices is represented by an intersectionof their segments.BUT the point of intersection is not an interior point of both linesegments.

Thus creating a "T" intersection.

ba

a

b

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 3 / 14

Wait! What about "V" or "X" intersections?

True, however, by T-segment properties:1 The point of intersection of two segments is not an interior point of

both segments , so segments cannot cross, "X".2 The point of intersection of two segments is not an endpoint of

both segments, so segments cannot meet in a "corner", "V".3 And, since parallel segments cannot intersect by reason 1, thus all

intersections take the "T" shape.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 4 / 14

T-segment representations

ba c

d

ba c

dnotice the loose ends, we’ll discuss that in a few slidesLet’s look at some properties of T-segment representations

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 5 / 14

T-segment representations

ba c

d

ba c

dnotice the loose ends, we’ll discuss that in a few slides

Let’s look at some properties of T-segment representations

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 5 / 14

T-segment representations

ba c

d

ba c

dnotice the loose ends, we’ll discuss that in a few slidesLet’s look at some properties of T-segment representations

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 5 / 14

Planar Graphs

A graph that can be drawn in such a way that no edges intersect orcross each other.

Examples of Planar Graphs

a b

d

c

e

a

c

d K4(notice the4 vertices)Butterfly Graph

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 6 / 14

Planar Graphs

A graph that can be drawn in such a way that no edges intersect orcross each other.

Examples of Planar Graphs

a b

d

c

e

a

c

d K4(notice the4 vertices)Butterfly Graph

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 6 / 14

Planar Graph or Not?

b ba b

d c

a

cd

b

notice the 4 vertices

We proved the Theorem: Every T-segment graph is a planar graph.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 7 / 14

Planar Graph or Not?

b ba b

d c

a

cd

b

notice the 4 vertices

We proved the Theorem: Every T-segment graph is a planar graph.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 7 / 14

Outerplanar Graphs

A graph that can be drawn so no vertex is totally surrounded by edges.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 8 / 14

Outerplanar Graphs

A graph that can be drawn so no vertex is totally surrounded by edges.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 8 / 14

What’s the big deal with the loose ends?

By representing graphs in T-Segments, three things happen:1. Every line segment contributes at most 2 endpoints and2. And every intersection uses up one endpoint.3. And every representation (with n > 1) has at least 3 loose ends

This leads us to find an edgebound formula for each graph, which ise ≤ 2n − 3.n is the number of vertices and e is the number of edges in a graph.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 9 / 14

What’s the big deal with the loose ends?

By representing graphs in T-Segments, three things happen:1. Every line segment contributes at most 2 endpoints and2. And every intersection uses up one endpoint.3. And every representation (with n > 1) has at least 3 loose ends

This leads us to find an edgebound formula for each graph, which ise ≤ 2n − 3.n is the number of vertices and e is the number of edges in a graph.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 9 / 14

Why Edgebound, part 1?

Consider this T-segment representation of a graph and its convex hull.If the convex hull is 2-D then we have a polygon with at least 3 corners.The corners come from loose ends.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 10 / 14

Why Edgebound, part 2?

Each line segment has 2 endpoints and uses one endpoint in eachintersection, so I have at most twice as many intersections (e) as Ihave segments (n) in the graph. Remember at least 3 of my endpointsare loose ends.

In our formula: e ≤ 2n − 3, the 3 represents the loose ends and the 2represents the 2 ends of each segment.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 11 / 14

Edgebound Example

This "elephant stool" graph cannot be drawn as a T-segmentrepresentation, because it does NOT meet the edgebound.

n = 8, e = 14here e 6≤ 2n − 3

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 12 / 14

Edgebound Example

This "elephant stool" graph cannot be drawn as a T-segmentrepresentation, because it does NOT meet the edgebound.

n = 8, e = 14here e 6≤ 2n − 3

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 12 / 14

Subgraphs

Take some of the vertices of G and some of the edges of G on thosevertices.Theorem: Every subgraph of a T-segment graph, is a T-segment graphtoo.

a

b

d

c A

B

C D

subgraphT-segmentsub-representation

a

b

d

c f

g

e

A

B

G

C

E

FD

graphT-segmentrepresentation

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 13 / 14

Kathleen’s ConjectureA graph G is a T-segment graph if and only if:

1 G is planar,2 2nG − 3 ≥ eG, and3 For any subgraph H of G, 2nH − 3 ≥ eH .

As many of you know, 4 weeks into our research we found an articlethat referenced this same formula, presented at a conference in 1993by a mathematician by the name of Thomassen......grrrrrr.

Later in 2004, two mathematicians, Hubert de Fraysseix and PatriceOssona de Mendez, published the same results. ......double grrrrrrr.

Melinda Kleinman (Willamette University) There And Back Again, A T-Segment Tale 8/10/12 14 / 14