arXiv subjects

Blerina Sinaimeri

Publications and source records attributed to Blerina Sinaimeri.

16 recordsLinked to original sources

arXiv2026

Disjoint covering of bipartite graphs with $s$-clubs

For a positive integer $s$, an $s$-club in a graph $G$ is a set of vertices inducing a subgraph with diameter at most $s$. As generalizations of cliques, $s$-clubs offer a flexible model for real-world networks. This paper addresses the problems of partitioning and disjoint covering of vertices with $s$-clubs on bipartite graphs. First we consider the $(k,s)$-PC problem where ask whether the vertices of $G$ can be partitioned into at most $k$ disjoint $s$-clubs. We prove that for any fixed $k \geq 2$ and for any fixed odd $s \geq 3$ or even $s\geq 8$, the $(k,s)$-PC problem is NP-complete even for bipartite graphs. Note that our NP-completeness result is stronger than the one in Abbas and Stewart (1999), as we assume that both $s$ and $k$ are constants and not part of the input. Additionally, we study the Maximum Disjoint $(t,s)$-Club Covering problem ($(t,s)$-MAX-DCC), which aims to find a collection of vertex-disjoint $(t,s)$-clubs (i.e. $s$-clubs with at least $t$ vertices) that covers the maximum number of vertices in $G$. We prove that it is NP-hard to achieve an approximation factor of $\frac{95}{94} $ for $(t,3)$-MAX-DCC for any fixed $t\geq 8$ and for $(t,2)$-MAX-DCC for any fixed $t\geq 5$ even for bipartite graphs. Previously, results were known only for $(3,2)$-MAX-DCC. Finally, we provide a polynomial-time algorithm for $(2,2)$-MAX-DCC resolving an open problem from Dondi \textit{et al.} (2019).

cs.CC↗

arXiv2026

Structural properties of distance-bounded phylogenetic reconciliation

Phylogenetic reconciliation seeks to explain host-symbiont co-evolution by mapping parasite trees onto host trees through events such as cospeciation, duplication, host switching, and loss. Finding an optimal reconciliation that ensures time feasibility is computationally hard when timing information is incomplete, and the complexity remains open when host switches are restricted by a fixed maximum distance $d$. While the case $d=2$ is known to be polynomial, larger values are unresolved. In this paper, we study the cases $d=3$ and $d=4$. We show that although arbitrarily large cycles may occur, it suffices to check only bounded-size cycles (we provide a complete list), provided the reconciliation satisfies acyclicity (i.e., time-feasibility) in a stronger sense. These results do not resolve the general complexity, but highlight structural properties that advance the understanding of distance-bounded reconciliations.

q-bio.PE↗

arXiv2024

Effects of graph operations on star pairwise compatibility graphs

A graph $G=(V,E)$ is defined as a star-$k$-PCG when it is possible to assign a positive real number weight $w$ to each vertex $V$, and define $k$ distinct intervals $I_1, I_2, \ldots I_k$, in such a way that there is an edge $uv$ in $E$ if and only if the sum of the weights of vertices $u$ and $v$ falls within the union of these intervals. The star-$k$-PCG class is connected to two significant categories of graphs, namely PCGs and multithreshold graphs. The star number of a graph $G$, is the smallest $k$ for which $G$ is a star-$k$-PCG. In this paper, we study the effects of various graph operations, such as the addition of twins, pendant vertices, universal vertices, or isolated vertices, on the star number of the graph resulting from these operations. As a direct application of our results, we determine the star number of lobster graphs and provide an upper bound for the star number of acyclic graphs.

cs.DM↗

arXiv2024

On Generalizations of Pairwise Compatibility Graphs

A graph $G$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and an interval $I$, 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 the interval $I$. Originating in phylogenetics, PCGs are closely connected to important graph classes like leaf-powers and multi-threshold graphs, widely applied in bioinformatics, especially in understanding evolutionary processes. In this paper we introduce two natural generalizations of the PCG class, namely $k$-OR-PCG and $k$-AND-PCG, which are the classes of graphs that can be expressed as union and intersection, respectively, of $k$ PCGs. These classes can be also described using the concepts of the covering number and the intersection dimension of a graph in relation to the PCG class. We investigate how the classes of OR-PCG and AND-PCG are related to PCGs, $k$-interval-PCGs and 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-PCGs, $k$-OR-PCG or $k$-AND-PCG classes. 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-PCGs class, proving that there is no finite $k$ for which the $k$-interval-PCG class contains all the graphs. This answers an open question of Ahmed and Rahman from 2017. Finally, using a Ramsey theory argument, we show that for any $k$, there exists graphs that are not in $k$-AND-PCG, and graphs that are not in $k$-OR-PCG.

math.CO↗

arXiv2024

On star-$k$-PCGs: Exploring class boundaries for small $k$ values

A graph $G=(V,E)$ is a star-$k$-PCG if there exists a weight function $w: V \rightarrow R^+$ and $k$ mutually exclusive intervals $I_1, I_2, \ldots I_k$, such that there is an edge $uv \in E$ if and only if $w(u)+w(v) \in \bigcup_i I_i$. These graphs are related to two important classes of graphs: PCGs and multithreshold graphs. It is known that for any graph $G$ there exists a $k$ such that $G$ is a star-$k$-PCG. Thus, for a given graph $G$ it is interesting to know which is the minimum $k$ such that $G$ is a star-$k$-PCG. We define this minimum $k$ as the star number of the graph, denoted by $γ(G)$. Here we investigate the star number of simple graph classes, such as graphs of small size, caterpillars, cycles and grids. Specifically, we determine the exact value of $γ(G)$ for all the graphs with at most 7 vertices. By doing so we show that the smallest graphs with star number 2 are only 4 and have exactly 5 vertices; the smallest graphs with star number 3 are only 3 and have exactly 7 vertices. Next, we provide a construction showing that the star number of caterpillars is one. Moreover, we show that the star number of cycles and two dimensional grid graphs is 2 and that the star number of $4$-dimensional grids is at least 3. Finally, we conclude with numerous open problems.

math.CO↗

arXiv2023

Cophylogeny Reconstruction Allowing for Multiple Associations Through Approximate Bayesian Computation

Phylogenetic tree reconciliation is employed for the examination of coevolution between host and symbiont species. An important concern is the requirement for dependable cost values when selecting event-based parsimonious reconciliation. Although certain approaches deduce event probabilities unique to each pair of host and symbiont trees, which can subsequently be converted into cost values, a significant limitation lies in their inability to model the invasion of diverse host species by the same symbiont species (termed as a spread event), which is believed to occur in symbiotic relationships. Invasions lead to the observation of multiple associations between symbionts and their hosts (indicating that a symbiont is no longer exclusive to a single host), which are incompatible with the existing methods of coevolution. We present AmoCoala, an enhanced version of the tool Coala, that provides a more realistic estimation of cophylogeny event probabilities for a given pair of host and symbiont trees, even in the presence of spread events. We expand the classical 4-event coevolutionary model to include 2 additional spread events (vertical and horizontal spreads) that lead to multiple associations. By incorporating spread events, our reconciliation model enables a more accurate consideration of multiple associations. This improvement enhances the precision of estimated cost sets, paving the way to a more reliable reconciliation of host and symbiont trees. Our results showcase that AmoCoala produces biologically plausible reconciliation scenarios, further emphasizing its effectiveness. The software is accessible at https://github.com/sinaimeri/AmoCoala

q-bio.QM↗

arXiv2023

A general framework for enumerating equivalence classes of solutions

When a problem has more than one solution, it is often important, depending on the underlying context, to enumerate (i.e., to list) them all. Even when the enumeration can be done in polynomial delay, that is, spending no more than polynomial time to go from one solution to the next, this can be costly as the number of solutions themselves may be huge, including sometimes exponential. Furthermore, depending on the application, many of these solutions can be considered equivalent. The problem of an efficient enumeration of the equivalence classes or of one representative per class (without generating all the solutions), although identified as a need in many areas, has been addressed only for very few specific cases. In this paper, we provide a general framework that solves this problem in polynomial delay for a wide variety of contexts, including optimization ones that can be addressed by dynamic programming algorithms, and for certain types of equivalence relations between solutions.

cs.DS↗

arXiv2022

On the domination number of $t$-constrained de Bruijn graphs

Motivated by the work on the domination number of directed de Bruijn graphs and some of its generalizations, in this paper we introduce a natural generalization of de Bruijn graphs (directed and undirected), namely $t$-constrained de Bruijn graphs, where $t$ is a positive integer, and then study the domination number of these graphs. Within the definition of $t$-constrained de Bruijn graphs, de Bruijn and Kautz graphs correspond to 1-constrained and 2-constrained de Bruijn graphs, respectively. This generalization inherits many structural properties of de Bruijn graphs and may have similar applications in interconnection networks or bioinformatics. We establish upper and lower bounds for the domination number on $t$-constrained de Bruijn graphs both in the directed and in the undirected case. These bounds are often very close and in some cases we are able to find the exact value.

math.CO↗

arXiv2019

String factorisations with maximum or minimum dimension

In this paper we consider two problems concerning string factorisation. Specifically given a string $w$ and an integer $k$ find a factorisation of $w$ where each factor has length bounded by $k$ and has the minimum (the FmD problem) or the maximum (the FMD problem) number of different factors. The FmD has been proved to be NP-hard even if $k=2$ in [9] and for this case we provide a $3/2$-approximation algorithm. The FMD problem, up to our knowledge has not been considered in the literature. We show that this problem is NP-hard for any $k\geq 3$. In view of this we propose a $2$-approximation algorithm (for any $k$) an exact exponential algorithm. We conclude with some open problems.

cs.DS↗

arXiv2017

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 $ϕ$ 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 $Rec(P,H,ϕ)$ 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 $Ψ$ of $Rec(P,H,ϕ)$ relative to $d$, i.e. an element $ψ_{med} \in Rec(P,H,ϕ)$ such that $$ \sum_{ψ' \in Ψ} d(ψ_{med},ψ') \le \sum_{ψ' \in Ψ} d(ψ,ψ') $$ holds for all $ψ\in Rec(P,H,ϕ)$? 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 $Rec(P,H,ϕ)$ can be super-exponential, it is still possible to compute a geometric median for a set $Ψ$ in $Rec(P,H,ϕ)$ 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).

q-bio.PE↗

arXiv2015

On Pairwise Compatibility of Some Graph (Super)Classes

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 `d' and `D' such that each leaf `u' of T is a node of V and the edge `(u,v) belongs to E' iff `d <= d_T(u, v) <= D' where d_T(u, v) is the sum of weights of the edges on the unique path from `u' to `v' in T. The main issue on these graphs consists in characterizing them. In this note we prove the inclusion in the PCG class of threshold tolerance graphs and the non-inclusion of a number of intersection graphs, such as disk and grid intersection graphs, circular arc and tolerance graphs. The non-inclusion of some superclasses (trapezoid, permutation and rectangle intersection graphs) follows.

cs.DM↗

arXiv2014

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.

cs.CC↗

arXiv2014

Navigating in a sea of repeats in RNA-seq without drowning

The main challenge in de novo assembly of NGS data is certainly to deal with repeats that are longer than the reads. This is particularly true for RNA- seq data, since coverage information cannot be used to flag repeated sequences, of which transposable elements are one of the main examples. Most transcriptome assemblers are based on de Bruijn graphs and have no clear and explicit model for repeats in RNA-seq data, relying instead on heuristics to deal with them. The results of this work are twofold. First, we introduce a formal model for repre- senting high copy number repeats in RNA-seq data and exploit its properties for inferring a combinatorial characteristic of repeat-associated subgraphs. We show that the problem of identifying in a de Bruijn graph a subgraph with this charac- teristic is NP-complete. In a second step, we show that in the specific case of a local assembly of alternative splicing (AS) events, we can implicitly avoid such subgraphs. In particular, we designed and implemented an algorithm to efficiently identify AS events that are not included in repeated regions. Finally, we validate our results using synthetic data. We also give an indication of the usefulness of our method on real data.

cs.DS↗

arXiv2012

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.

cs.DM↗

arXiv2011

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.

cs.DM↗

arXiv2007

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".

math.CO↗