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

Relating threshold tolerance graphs to other graph classes

A graph G=(V, E) is a threshold tolerance if it is possible to associate weights and tolerances with each node of G so that two nodes are adjacent exactly when the sum of their weights exceeds either one of their tolerances. Threshold tolerance graphs are a special case of the well-known class of tolerance graphs and generalize the class of threshold graphs which are also extensively studied in literature. In this note we relate the threshold tolerance graphs with other important graph classes. In particular we show that threshold tolerance graphs are a proper subclass of co-strongly chordal graphs and strictly include the class of co-interval graphs. To this purpose, we exploit the relation with another graph class, min leaf power graphs (mLPGs)

On dynamic threshold graphs and related classes

This paper deals with the well known classes of threshold and difference graphs, both characterized by separators, i.e. node weight functions and thresholds. We design an efficient algorithm to find the minimum separator, and we show how to maintain minimum its value when the input (threshold or difference) graph is fully dynamic, i.e. edges/nodes are inserted/removed. Moreover, exploiting the data structure used for maintaining the minimality of the separator, we study the disjoint union and the join of two threshold graphs, showing that the resulting graphs are threshold signed graphs, i.e. a superclass of both threshold and difference graphs. Finally, we consider the complement operation on all the three introduced classes of graphs. All these operations produce in output the modified graph in terms of their separator and require time linear w.r.t. the number of different degrees. We observe that recomputing from scratch the separator would run either in linear (for threshold and difference graphs) or quadratic (for threshold signed graphs) time w.r.t. the number of nodes of the graph

On relaxing the constraints in pairwise compatibility graphs

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 $u \in V$ and there is an edge $(u,v) \in E$ if and only if $d_{min} \leq d_T (l_u, l_v) \leq d_{max}$ 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 \dmin=0 (LPG) and \dmax=+\infty (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.Comment: 12 pages, 7 figure

Dynamically mantaining minimal integral separator for Threshold and Difference Graphs

This paper deals with the well known classes of threshold and difference graphs, both characterized by separators, i.e. node weight functions and thresholds. We show how to maintain minimum the value of the separator when the input (threshold or difference) graph is fully dynamic, i.e. edges/nodes are inserted/removed. Moreover, exploiting the data structure used for maintaining the minimality of the separator, we handle the operations of disjoint union and join of two threshold graphs. © Springer International Publishing Switzerland 2016

All graphs with at most seven vertices are Pairwise Compatibility Graphs

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 $u \in V$ and there is an edge $(u,v) \in E$ if and only if $d_{min} \leq d_{T,w} (l_u, l_v) \leq d_{max}$ where $d_{T,w} (l_u, l_v)$ is the sum of the weights of the edges on the unique path from $l_u$ to $l_v$ in $T$. In this 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 $W_7$ are PCGs of a particular structure of a tree: a centipede.Comment: 8 pages, 2 figure

A simple linear time algorithm for the locally connected spanning tree problem on maximal planar chordal graphs

A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that, for each node, its neighborhood in T induces a connected sub- graph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a simple linear time algorithm that, given a maximal planar chordal graph, determines in linear time whether it contains an LCST or not, and produces one if it exists. We give an anal- ogous result for the case when the input graph is a maximal outerplanar graph

Extracting few representative reconciliations with Host-Switches (Extended Abstract)

Phylogenetic tree reconciliation is the approach commonly used to in- vestigate the coevolution of sets of organisms such as hosts and symbionts. Given a phylogenetic tree for each such set, respectively denoted by H and S, together with a mapping φ of the leaves of S to the leaves of H, a reconciliation is a mapping ρ of the internal vertices of S to the vertices of H which extends φ with some constraints. Given a cost for each reconciliation, a huge number of most parsimonious ones are possible, even exponential in the dimension of the trees. Without further information, any biological interpretation of the underlying coevolution would require that all optimal solutions are enumerated and examined. The latter is however impossible without pro- viding some sort of high level view of the situation. One approach would be to extract a small number of representatives, based on some notion of similarity or of equivalence between the reconciliations. In this paper, we define two equivalence relations that allow one to identify many reconciliations with a single one, thereby reducing their number. Extensive experiments indicate that the number of output solutions greatly decreases in general. By how much clearly depends on the constraints that are given as input