All Planar Graphs of 4 Vertices are Contact Cover Graph?
Md. Taufique Hussain (0805050), Shohedul Hasan(0805059),
Supervisor: Md. Saidur Rahman
Problem Definition
Motivation and Challenge
• Any geometric embedding of any three or less vertex
planar graph can be represented as Contact Cover
Graph.
• There is an embedding of a six vertex graph which
can’t be represented as Contact Cover Graph.
A geometric embedding M of
a 3 vertex planar graph
Application
Problems related to these graphs arise in many application areas
like –
• Wireless Communication Networks [2]
• Facility Locations [3]
A contact cover graph representation G
of embedding M
Let M be an embedding of a planar graph G = (V, E). M can be represented
as a contact cover graph if there exists |V| circles in the plane each of
which contains only one of the vertices in the area enclosed by them, they
touch each other only if the respective vertices contained by them have an
edge between them and they never overlap with each other.
• But it is still not proved if all planar graphs of 4 and 5
vertices can be represented as Contact Cover Graph
or not.
Literatures
Outcome
• So far we have discovered 14 classes of different embedding
of 4 vertex planar graphs depending different edge length and
Different angle between edges. Three of them are -
[1] Nieves Atienza, Natalia de Castro et al.
Cover Contact Graphs
[2] B. N. Clark, C. J. Colbourn, and D. S. Johnson.
Unit disk graphs. Discrete Mathematics, 86:165 177, 1990.
• Among these 14, 11 of the classes can be represented by CCG
In one way or another in some algorithm. Three of the
representations are -
• Still 3 more classes are yet to be resolved. And it needs to be
proved that no other class exists except these 14 classes.
[3] J.-M. Robert and G. T. Toussaint. Computational
geometry and facility location. In International
Conference on Operations Research and Management
Sciences, pages B-1-B-19, Manila, Philippines, 1990.
Department of Computer Science and Engineering (CSE), BUET