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.


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


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.


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



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



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


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


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.


Outerplanar Graphs

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


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.


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.


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.


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


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


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.


There And Back Again, A T-Segment...

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