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Erschienen in:
2012 | OriginalPaper | Buchkapitel
On Relaxing the Constraints in Pairwise Compatibility Graphs
verfasst von : Tiziana Calamoneri, Rossella Petreschi, Blerina Sinaimeri
Erschienen in: WALCOM: Algorithms and Computation
Verlag: Springer Berlin Heidelberg
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A graph
G
is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree
T
and two non-negative real numbers
d
min
and
d
max
such that each leaf
l
u
of
T
corresponds to a vertex
u
∈
V
and there is an edge (
u
,
v
) ∈
E
if and only if
d
min
≤
d
T
(
l
u
,
l
v
) ≤
d
max
where
d
T
(
l
u
,
l
v
) is the sum of the weights of the edges on the unique path from
l
u
to
l
v
in
T
. In this paper we analyze the class of PCG in relation with two particular subclasses resulting from the the cases where
d
min
= 0 (LPG) and
d
max
= + ∞ (mLPG). In particular, we show that the union of LPG and mLPG does not coincide with the whole class PCG, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, as the graphs we deal with belong to the more general class of split matrogenic graphs, we focus on this class of graphs for which we try to establish the membership to the PCG class.