1,749 research outputs found
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
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
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
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
On strongly chordal graphs that are not leaf powers
A common task in phylogenetics is to find an evolutionary tree representing
proximity relationships between species. This motivates the notion of leaf
powers: a graph G = (V, E) is a leaf power if there exist a tree T on leafset V
and a threshold k such that uv is an edge if and only if the distance between u
and v in T is at most k. Characterizing leaf powers is a challenging open
problem, along with determining the complexity of their recognition. This is in
part due to the fact that few graphs are known to not be leaf powers, as such
graphs are difficult to construct. Recently, Nevries and Rosenke asked if leaf
powers could be characterized by strong chordality and a finite set of
forbidden subgraphs.
In this paper, we provide a negative answer to this question, by exhibiting
an infinite family \G of (minimal) strongly chordal graphs that are not leaf
powers. During the process, we establish a connection between leaf powers,
alternating cycles and quartet compatibility. We also show that deciding if a
chordal graph is \G-free is NP-complete, which may provide insight on the
complexity of the leaf power recognition problem
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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