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

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)

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

Computing an Evolutionary Ordering is Hard

We prove that computing an evolutionary ordering of a family of sets, i.e. an ordering where each set intersects with --but is not included in-- the union earlier sets, is NP-hard

L(2,1)-labeling of oriented planar graphs

The L(2, 1)-labeling of a digraph D is a function l from the vertex set of D to the set of all nonnegative integers such that vertical bar l(x) - l(y)vertical bar >= 2 if x and y are at distance 1, and l(x) not equal l(y) if x and y are at distance 2, where the distance from vertex x to vertex y is the length of a shortest dipath from x to y. The minimum over all the L(2, 1)-labelings of D of the maximum used label is denoted (lambda) over right arrow (D). If C is a class of digraphs, the maximum (lambda) over right arrow (D), over all D is an element of C is denoted (lambda) over right arrow (C). In this paper we study the L(2, 1)-labeling problem on oriented planar graphs providing some upper bounds on (lambda) over right arrow. Then we focus on some specific subclasses of oriented planar graphs, improving the previous general bounds. Namely, for oriented prisms we compute the exact value of (lambda) over right arrow, while for oriented Halin graphs and cacti we provide very close upper and lower bounds for (lambda) over right arrow. (c) 2012 Elsevier B.V. All rights reserved

L(2,1)-labeling of oriented planar graphs

The L(2, 1)-labeling of a digraph D is a function l from the vertex set of D to the set of all nonnegative integers such that vertical bar l(x) - l(y)vertical bar >= 2 if x and y are at distance 1, and l(x) not equal l(y) if x and y are at distance 2, where the distance from vertex x to vertex y is the length of a shortest dipath from x to y. The minimum over all the L(2, 1)-labelings of D of the maximum used label is denoted (lambda) over right arrow (D). If C is a class of digraphs, the maximum (lambda) over right arrow (D), over all D is an element of C is denoted (lambda) over right arrow (C). In this paper we study the L(2, 1)-labeling problem on oriented planar graphs providing some upper bounds on (lambda) over right arrow. Then we focus on some specific subclasses of oriented planar graphs, improving the previous general bounds. Namely, for oriented prisms we compute the exact value of (lambda) over right arrow, while for oriented Halin graphs and cacti we provide very close upper and lower bounds for (lambda) over right arrow. (c) 2012 Elsevier B.V. All rights reserved

On types of growth for graph-different permutations

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity "within a given type".Comment: 14 pages+title pag

Geometric medians in reconciliation spaces

In evolutionary biology, it is common to study how various entities evolve together, for example, how parasites coevolve with their host, or genes with their species. Coevolution is commonly modelled by considering certain maps or reconciliations from one evolutionary tree $P$ to another $H$, all of which induce the same map $\phi$ between the leaf-sets of $P$ and $H$ (corresponding to present-day associations). Recently, there has been much interest in studying spaces of reconciliations, which arise by defining some metric $d$ on the set $Rec(P,H,\phi)$ of all possible reconciliations between $P$ and $H$. In this paper, we study the following question: How do we compute a geometric median for a given subset $\Psi$ of $Rec(P,H,\phi)$ relative to $d$, i.e. an element $\psi_{med} \in Rec(P,H,\phi)$ such that $$\sum_{\psi' \in \Psi} d(\psi_{med},\psi') \le \sum_{\psi' \in \Psi} d(\psi,\psi')$$ holds for all $\psi \in Rec(P,H,\phi)$? For a model where so-called host-switches or transfers are not allowed, and for a commonly used metric $d$ called the edit-distance, we show that although the cardinality of $Rec(P,H,\phi)$ can be super-exponential, it is still possible to compute a geometric median for a set $\Psi$ in $Rec(P,H,\phi)$ in polynomial time. We expect that this result could be useful for computing a summary or consensus for a set of reconciliations (e.g. for a set of suboptimal reconciliations).Comment: 12 pages, 1 figur

On Generalizations of Pairwise Compatibility Graphs

A graph $G$ 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 $\{ x, y \}$ in $G$ if and only if the weight of the path in the tree connecting $x$ and $y$ 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 $k$ 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 $k$ for which an arbitrary graph $G$ 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 $k$, there exists a bipartite graph that is not in the k-interval-PCG class, proving that there is no finite $k$ for which the k-interval-PCG class contains all the graphs. Finally, we use a Ramsey theory argument to show that for any $k$, there exist graphs that are not in k-AND-PCG, and graphs that are not in k-OR-PCG