arXiv subjects

Annachiara Korchmaros

Publications and source records attributed to Annachiara Korchmaros.

14 recordsLinked to original sources

arXiv2025

Forbidden configurations and dominating bicliques in undirected 2-quasi best match graphs

2-quasi best match graphs (2-qBMGs) are directed graphs that capture a notion of close relatedness in phylogenetics. Here, we investigate the undirected underlying graph of a 2-qBMG (un-2qBMG) and show that they contain neither a path $P_l$ nor a cycle $C_l$ of length $l\geq 6$ as an induced subgraph. This property guarantees the existence of specific vertex decompositions with dominating bicliques that provide further insights into their structure.

math.CO↗

arXiv2025

Characterizations of undirected 2-quasi best match graphs

Bipartite best match graphs (BMG) and their generalizations arise in mathematical phylogenetics as combinatorial models describing evolutionary relationships among related genes in a pair of species. In this work, we characterize the class of \emph{undirected 2-quasi-BMGs} (un2qBMGs), which form a proper subclass of the $P_6$-free chordal bipartite graphs. We show that un2qBMGs are exactly the class of bipartite graphs free of $P_6$, $C_6$, and the eight-vertex Sunlet$_4$ graph. Equivalently, a bipartite graph $G$ is un2qBMG if and only if every connected induced subgraph contains a ``heart-vertex'' which is adjacent to all the vertices of the opposite color. We further provide a $O(|V(G)|^3)$ algorithm for the recognition of un2qBMGs that, in the affirmative case, constructs a labeled rooted tree that ``explains'' $G$. Finally, since un2qBMGs coincide with the $(P_6,C_6)$-free bi-cographs, they can also be recognized in linear time.

math.CO↗

arXiv2025

Computing phylogenetic invariants for time-reversible models: from TN93 to its submodels

Phylogenetic invariants are equations that vanish on algebraic varieties associated with Markov processes that model molecular substitutions on phylogenetic trees. For practical applications, it is essential to understand these equations across a wide range of substitution models. Recent work has shown that, for equivariant models, phylogenetic invariants can be derived from those of the general Markov model by restricting to the linear space defined by the model (namely, the space of mixtures of distributions on the model). Following this philosophy, we describe the space of mixtures and phylogenetic invariants for time-reversible models that are not equivariant. Specifically, we study two submodels of the Tamura-Nei nucleotide substitution model (Felsenstein 81 and 84) using an orthogonal change of basis recently introduced for algebraic time-reversible models. For tripods, we prove that the algebraic variety of each submodel coincides with the variety of Tamura-Nei intersected with the linear space of the submodel. In the case of quartets, we show that it is an irreducible component of this intersection. Moreover, we demonstrate that it suffices to consider only the binomial equations defining the linear space, which correspond to the natural symmetries of the model in the new coordinates. For each submodel, we explicitly provide equations defining a local complete intersection that characterizes the phylogenetic variety on a dense open subset containing the biologically relevant points.

q-bio.PE↗

arXiv2025

Automorphisms and quotients of 2-colored quasi best match graphs

2-colored quasi best match graphs (2-qBMGs) are directed graphs that arose in phylogenetics. Investigations of 2-qBMGs have mostly focused on computational issues. However, 2-qBMGs also have relevant properties for structural graph theory; in particular, their undirected underlying graph is free from induced paths and cycles of size at least 6. In this paper, results on the structure of the automorphism groups of 2-qBMGs are obtained, which shows how to construct 2-qBMGs with large automorphism groups.

math.CO↗

arXiv2024

Forbidden paths and cycles in the undirected underlying graph of a 2-quasi best match graph

The undirected underlying graph of a 2-quasi best match graph (2-qBMG) is proven not to contain any induced graph isomorphic to $P_6$ or $C_6$. This new feature allows for the investigation of 2-BMGs further by exploiting the numerous known results on $P_6$ and $C_6$ free graphs together with the available polynomial algorithms developed for their studies. In this direction, there are also some new contributions about dominating bicliques and certain vertex decompositions of the undirected underlying graph of a 2-qBMG.

math.CO↗

arXiv2024

3D genome reconstruction from partially phased Hi-C data

The 3-dimensional (3D) structure of the genome is of significant importance for many cellular processes. In this paper, we study the problem of reconstructing the 3D structure of chromosomes from Hi-C data of diploid organisms, which poses additional challenges compared to the better-studied haploid setting. With the help of techniques from algebraic geometry, we prove that a small amount of phased data is sufficient to ensure finite identifiability, both for noiseless and noisy data. In the light of these results, we propose a new 3D reconstruction method based on semidefinite programming, paired with numerical algebraic geometry and local optimization. The performance of this method is tested on several simulated datasets under different noise levels and with different amounts of phased data. We also apply it to a real dataset from mouse X chromosomes, and we are then able to recover previously known structural features.

q-bio.GN↗

arXiv2024

Configurations in the Euclidean space related to the 3D genome reconstruction problem from partially phased data

A motivation for studying the following problems comes from applications to Biology; see \cite{cifuentes20233d}. In the $3$-dimensional Euclidean space ${\bf{E}}^3$, fix six pairwise distinct points \begin{equation*} \label{eqA} \begin{array}{ccc} A=(a_1,a_2,a_3), & B=(b_1,b_2,b_3), & C=(c_1,c_2,c_3), \\ D=(d_1,d_2,d_3), & E=(e_1,e_2,e_3), & F=(f_1,f_2,f_3) \end{array} \end{equation*} together with two further points $X^*=(x_1^*,x_2^*,x_3^*)$ and $Y^*=(y_1^*,y_2^*,y_3^*)$ in $\mathbf{E}^3$. We aim to show that System $(*)$ consisting of the following six equations in the unknowns $X=(x_1,x_2,x_3)$ and $Y=(y_1,y_2,y_3)$ \begin{equation} \label{egy} \frac{1}{\|X-T\|^2} +\frac{1}{\|Y-T\|^2}=\frac{1}{\|X^*-T\|^2} +\frac{1}{\|Y^*-T\|^2}, \quad T\in\{A,B,\ldots,F\}. \end{equation} has only finitely many solutions provided that both of the following two conditions are satisfied: (i) no four of the fixed points $A,B,C,D,E,F$ are coplanar; (ii) no four of the six spheres of center $T$ and radius $1/\sqrt{k_T}$ with \begin{equation} \label{kxy} k_T=\frac{1}{\|X^*-T\|^2} +\frac{1}{\|Y^*-T\|^2} \end{equation} share a common point in ${\bf{E}}^3$. Furthermore, we exhibit configurations $ABCDEFX^*Y^*$, showing that (i) is also necessary. This result is an improvement on \cite[Theorem 1]{cifuentes20233d} where the finiteness of solutions of System $(*)$ is only ensured for sufficiently generic choices of the points $A,B,\ldots,F,X^*,Y^*.$ We also show if System $(*)$ has finitely many solutions and System $(*)$ extended with $T\in\{A,B,\ldots,F,G\}$ has some solutions other than $(X^*,Y^*)$ and $(Y^*,X^*)$ then $G$ lies on an explicitly given affine variety $W\subsetneqq \mathbb{R}^3$ only depending on $\{A,B,\ldots,F\}$. This result proves the \cite[Conjecture 1]{cifuentes20233d}.

math.AG↗

arXiv2023

Quasi-Best Match Graphs

Quasi-best match graphs (qBMGs) are a hereditary class of directed, properly vertex-colored graphs. They arise naturally in mathematical phylogenetics as a generalization of best match graphs, which formalize the notion of evolutionary closest relatedness of genes (vertices) in multiple species (vertex colors). They are explained by rooted trees whose leaves correspond to vertices. In contrast to BMGs, qBMGs represent only best matches at a restricted phylogenetic distance. We provide characterizations of qBMGs that give rise to polynomial-time recognition algorithms and identify the BMGs as the qBMGs that are color-sink-free. Furthermore, two-colored qBMGs are characterized as directed graphs satisfying three simple local conditions, two of which have appeared previously, namely bi-transitivity in the sense of Das et al. (2021) and a hierarchy-like structure of out-neighborhoods, i.e., $N(x)\cap N(y)\in\{N(x),N(y),\emptyset\}$ for any two vertices $x$ and $y$. Further results characterize qBMGs that can be explained by binary phylogenetic trees.

math.CO↗

arXiv2020

The Structure of 2-Colored Best Match Graphs

Recent investigations in computational biology have focused on a family of 2-colored digraphs, called 2-colored best match graphs, which naturally arise from rooted phylogenetic trees. Actually the defining properties of such graphs are unusual, and a natural question is whether they also have properties which well fit in structural graph theory. In this paper, we prove that some underlying oriented bipartite graphs of a 2-colored best match graph are acyclic and we point out that the arising topological ordering can efficiently be used for constructing new families of 2-colored best match graphs.

math.CO↗

arXiv2020

Circles and Paths in 2-Colored Best Match Graphs

Recent investigations in computational biology focus on a family of 2-colored digraphs, called 2-colored best match graphs, which naturally arise from rooted phylogenetic trees. Actually the defining properties of such graphs are unexpectedly unusual in graph theory, and they were established only recently after the discovery of their links to evolutionary relatedness via phylogenetic trees. In this paper several results are obtained on 2-colored best match graphs which well fit in the mainstream of graph theory.

math.CO↗

arXiv2015

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

This paper deals with the Cayley graph $\mathrm{Cay}(\mathrm{Sym}_n,T_n),$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that Aut$(\mathrm{Cay}(\mathrm{Sym}_n,T_n))$ is the product of the left translation group by a dihedral group $\mathsf{D}_{n+1}$ of order $2(n+1)$. The proof uses several properties of the subgraph $Γ$ of $\mathrm{Cay}(\mathrm{Sym}_n,T_n)$ induced by the set $T_n$. In particular, $Γ$ is a $2(n-2)$-regular graph whose automorphism group is $\mathsf{D}_{n+1},$ $Γ$ has as many as $n+1$ maximal cliques of size $2,$ and its subgraph $Γ(V)$ whose vertices are those in these cliques is a $3$-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of $\mathsf{D}_{n+1}$ of order $n+1$ with regular Cayley maps on $\mathrm{Sym}_n$ is also discussed. It is shown that the product of the left translation group by the latter group can be obtained as the automorphism group of a non-$t$-balanced regular Cayley map on $\mathrm{Sym}_n$.

math.CO↗

arXiv2015

Combinatorial properties of block transpositions on Symmetric groups

A major problem in the study of combinatorial aspects of permutation groups is to determine the distances in the symmetric group $\Sym_n$ with respect to a generator set. One well-known such a case is when the generator set $S_n$ consists of block transpositions. It should be noted that "the block transposition distance of a permutation" is the distance of the permutation from the identity permutation in the Cayley graph $\Cay$, and "sorting a permutation by block transpositions" is equivalent to finding shortest paths in $\Cay$. The original results in our thesis concern the lower and upper bounds on the block transpositions diameter of $\Sym_n$ with respect to $S_n$ and the automorphism group $\Aut(\Cay)$. A significant contribution is to show how from the toric equivalence can be obtained bijective maps on $\Sym_n$ that we call \emph{toric maps}. Using the properties of the toric maps, we discuss the role of the invariance principle of the block transposition distance within toric classes in the proof of the Eriksson bound. Furthermore, we prove that $\Aut(\Cay)$ is the product of the right translation group by $\textsf{N}\rtimes\textsf{D}_{n+1}$, where $\textsf{N}$ is the subgroup fixing $S_n$ elementwise, and $\textsf{D}_{n+1}$ is a dihedral group whose maximal cyclic subgroup is generated by the toric maps. Computer aided computation supports our conjecture that $\textsf{N}$ is trivial. Also, we prove that the subgraph $Γ$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whose automorphism group is $\textsf{D}_{n+1}$. We show some aspects of $\Cay$, notably $Γ$ has as many as $n+1$ maximal cliques of size $2$, its subgraph $Γ(V)$ whose vertices are those in these cliques is a $3$-regular Hamiltonian graph, and $\textsf{D}_{n+1}$ acts faithfully on $V$ as a vertex regular automorphism group.

math.CO↗

arXiv2015

Cayley graph on symmetric groups with generating block transposition sets

This paper deals with the Cayley graph $\Cay,$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. We prove that ${\rm{Aut}}(\Cay)$ is the product of the right translation group by $\textsf{N}\rtimes \textsf{D}_{n+1},$ where $\textsf{N}$ is the subgroup fixing $S_n$ element-wise and $\textsf{D}_{n+1}$ is a dihedral group of order $2(n+1)$. We conjecture that $\textsf{N}$ is trivial. We also prove that the subgraph $Γ$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whose automorphism group is $\textsf{D}_{n+1}$. Furthermore, $Γ$ has as many as $n+1$ maximum cliques of size $2.$ Also, its subgraph $Γ(V)$ whose vertices are those in these cliques is a $3$-regular, Hamiltonian, and vertex-transitive graph.

math.CO↗

arXiv2014

Upper bound on the block transposition diameter of the symmetric group

Given a generator set $S$ of the symmetric group ${\rm{Sym}}_n$, every permutation $π\in {\rm{Sym}_n}$ is a word (product of elements) of $S$. A positive integer $d(π)$ is associated with each $π\in{\rm{Sym}_n}$ taking the length of the shortest such word, and the $S$-diameter $d(S)$ is the maximum value of $d(π)$ with $π$ ranging over ${\rm{Sym_n}}$. The distance $d(π,ν)$ of two permutations $π,ν$ defined by $d(ν^{-1}\circπ)$ satisfies the axioms of a metric space. In this paper we consider the case where $S$ consists of all block transpositions of ${\rm{Sym_n}}$ and call $d(π)$ the block transposition distance of $π$. A strong motivation for the study of this special case comes from investigations of large-scale mutations of genome, where determining $d(π)$ is known as sorting the permutation $π$ by block transpositions. In the papers on this subject, toric equivalence classes often play a crucial role since $d(π)=d(ν)$ when $π$ and $ν$ are torically equivalent. A proof of this result can be found in the (unpublished) Hausen's Ph.D Dissertation thesis; see \cite{Ha}. Our main contribution is to obtain a bijective map on ${\rm{Sym}_n}$ from the toric equivalence that leaves the distances invariant. Using the properties of this map, we give an alternative proof of Hausen's result which actually fills a gap in the proof of the upper bound on $d(S)$ due to Eriksson and his coworkers; see \cite{EE}. We also revisit the proof of the key lemma \cite[Lemma 5,1]{EE}, giving more details and filling some gaps.

math.CO↗