3 research outputs found
Hierarchical Colorings of Cographs
Cographs are exactly hereditarily well-colored graphs, i.e., the graphs for
which a greedy coloring of every induced subgraph uses only the minimally
necessary number of colors . In recent work on reciprocal best match
graphs so-called hierarchically coloring play an important role. Here we show
that greedy colorings are a special case of hierarchical coloring, which also
require no more than colors
Hierarchical and Modularly-Minimal Vertex Colorings
Cographs are exactly the hereditarily well-colored graphs, i.e., the graphs
for which a greedy vertex coloring of every induced subgraph uses only the
minimally necessary number of colors . We show that greedy colorings
are a special case of the more general hierarchical vertex colorings, which
recently were introduced in phylogenetic combinatorics. Replacing cotrees by
modular decomposition trees generalizes the concept of hierarchical colorings
to arbitrary graphs. We show that every graph has a modularly-minimal coloring
satisfying for every strong module of .
This, in particular, shows that modularly-minimal colorings provide a useful
device to design efficient coloring algorithms for certain hereditary graph
classes. For cographs, the hierarchical colorings coincide with the
modularly-minimal coloring. As a by-product, we obtain a simple linear-time
algorithm to compute a modularly-minimal coloring of -sparse graphs.Comment: arXiv admin note: text overlap with arXiv:1906.1003
Complexity of Modification Problems for Reciprocal Best Match Graphs
Reciprocal best match graphs (RBMGs) are vertex colored graphs whose vertices
represent genes and the colors the species where the genes reside. Edges
identify pairs of genes that are most closely related with respect to an
underlying evolutionary tree. In practical applications this tree is unknown
and the edges of the RBMGs are inferred by quantifying sequence similarity. Due
to noise in the data, these empirically determined graphs in general violate
the condition of being a ``biologically feasible'' RBMG. Therefore, it is of
practical interest in computational biology to correct the initial estimate.
Here we consider deletion (remove at most edges) and editing (add or delete
at most edges) problems. We show that the decision version of the deletion
and editing problem to obtain RBMGs from vertex colored graphs is NP-hard.
Using known results for the so-called bicluster editing, we show that the RBMG
editing problem for -colored graphs is fixed-parameter tractable.
A restricted class of RBMGs appears in the context of orthology detection.
These are cographs with a specific type of vertex coloring known as
hierarchical coloring. We show that the decision problem of modifying a
vertex-colored graph (either by edge-deletion or editing) into an RBMG with
cograph structure or, equivalently, to an hierarchically colored cograph is
NP-complete