360 research outputs found
Pairwise Compatibility Graphs (Invited Talk)
Pairwise Compatibility Graphs (PCG) are graphs introduced in relation to the biological problem of reconstructing phylogenetic trees. Without demanding to be exhaustive, in this note we take a quick look at what is known in the literature for these graphs. The evolutionary history of a set of organisms is usually represented by a tree-like structure called phylogenetic tree, where the leaves are the known species and the internal nodes are the possible ancestors that might have led, through evolution, to this set of species. Edges are evolutionary relationships between species, while the edge weights represent evolutionary distances among species (evolutionary times). The phylogenetic tree reconstruction problem consists in finding a fully labeled phylogenetic tree that'best' explains the evolution of given species, where'best' means that it optimizes a specific target function. Tree reconstruction problem is proved to be NP-hard under many criteria of optimality, so the performance of the heuristics for this problem is usually experimentally evaluated by comparing the output trees with the partial trees that are unanimously recognized as sure by biologists. But real data consist of a huge number of species, and it is unfeasible to compare trees with such a number of leaves, so it is common to exploit sample techniques. The idea is to find efficient ways to sample subsets of species from a large set in order to test the heuristics on the smaller sub-trees induced by the sample. The constraints on the sample attempt to ensure that the behavior of the heuristics will not be biased by the fact it is applied on the sample instead of on the whole tree. Since very close or very distant taxa can create problems for phylogenetic reconstruction heuristics [9], the following definition of Pairwise Compatibility Graphs [12] appears natura
Graphs that are not pairwise compatible: A new proof technique (extended abstract)
A graph G = (V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dminand dmax, dmin≤ dmax, such that each node u∈V is uniquely associated to a leaf of T and there is an edge (u, v) ∈ E if and only if dmin≤ dT(u, v) ≤ dmax, where dT(u, v) is the sum of the weights of the edges on the unique path PT(u, v) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. Despite numerous efforts, a complete characterization of the PCG class is not known yet. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. We demonstrate our technique by showing many graph classes that do not lie in PCG. As a side effect, we show a not pairwise compatibility planar graph with 8 nodes (i.e. C28), so improving the previously known result concerning the smallest planar graph known not to be PCG
All graphs with at most seven vertices are Pairwise Compatibility Graphs
A graph is called a pairwise compatibility graph (PCG) if there exists an
edge-weighted tree and two non-negative real numbers and
such that each leaf of corresponds to a vertex
and there is an edge if and only if where is the sum of the weights of the
edges on the unique path from to in .
In this note, we show that all the graphs with at most seven vertices are
PCGs. In particular all these graphs except for the wheel on 7 vertices
are PCGs of a particular structure of a tree: a centipede.Comment: 8 pages, 2 figure
On Generalizations of Pairwise Compatibility Graphs
A graph is a PCG if there exists an edge-weighted tree such that each
leaf of the tree is a vertex of the graph, and there is an edge in
if and only if the weight of the path in the tree connecting and
lies within a given interval. PCGs have different applications in phylogenetics
and have been lately generalized to multi-interval-PCGs. In this paper we
define two new generalizations of the PCG class, namely k-OR-PCGs and
k-AND-PCGs, that are the classes of graphs that can be expressed as union and
intersection, respectively, of PCGs. The problems we consider can be also
described in terms of the \emph{covering number} and the \emph{intersection
dimension} of a graph with respect to the PCG class. In this paper we
investigate how the classes of PCG, multi-interval-PCG, OR-PCG and AND-PCG are
related to each other and to other graph classes known in the literature. In
particular, we provide upper bounds on the minimum for which an arbitrary
graph belongs to k-interval-PCG, k-OR-PCG and k-AND-PCG classes.
Furthermore, for particular graph classes, we improve these general bounds.
Moreover, we show that, for every integer , there exists a bipartite graph
that is not in the k-interval-PCG class, proving that there is no finite
for which the k-interval-PCG class contains all the graphs. Finally, we use a
Ramsey theory argument to show that for any , there exist graphs that are
not in k-AND-PCG, and graphs that are not in k-OR-PCG
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