2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 8792
author:  Martin Charles Golumbic, Marina Lipshteyn and Michal Stern 

title:  Representations of Edge Intersection Graphs of Paths in a Tree 
keywords:  Paths of a tree, Intersection graphs, Weakly chordal graphs, Coloring, EPTgraphs 
abstract: 
Let
P
T
. The edge intersection graph of
P
EPT(
, has vertex set that corresponds to the members of
P
)
P
P
T
. An undirected graph
G
is called an edge intersection graph of paths in a
tree, if
G
=
EPT(
for some
P
)
P
T
. The EPT graphs are useful in network applications.
Scheduling undirected calls in a tree or assigning
wavelengths to virtual connections in an optical tree
network are equivalent to coloring its EPT graph. It is
known that recognition and coloring of EPT graphs are
NPcomplete problems. However, the EPT graphs restricted to
host trees of vertex degree 3 are precisely the chordal EPT
graphs, and therefore can be colored in polynomial time
complexity. We prove a new analogous result that weakly
chordal EPT graphs are precisely the EPT graphs with host
tree restricted to degree 4. This also implies that the
coloring of the edge intersection graph of paths in a
degree 4 tree is polynomial. We raise a number of
intriguing conjectures regarding related families of
graphs.

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reference:  Martin Charles Golumbic and Marina Lipshteyn and Michal Stern (2005), Representations of Edge Intersection Graphs of Paths in a Tree, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 8792 
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