arXiv subjects

Ana Silva

Publications and source records attributed to Ana Silva.

At least 37 records · Page 2Linked to original sources

arXiv2021

Königsberg Sightseeing: Eulerian Walks in Temporal Graphs

An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph $G$ at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. What if Euler had to take a bus? In a temporal graph $(G,λ)$, with $λ: E(G)\to 2^{[τ]}$, an edge $e\in E(G)$ is available only at the times specified by $λ(e)\subseteq [τ]$, in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this scenario, even though several translations of Eulerian trails and walks are possible in temporal terms, only a very particular variation has been exploited in the literature, specifically for infinite dynamic networks (Orlin, 1984). In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether $(G,λ)$ has a temporal walk or trail is polynomial, while deciding whether it has a local trail is NP-complete even if it has lifetime~2. In contrast, in the general case, solving any of these problems is NP-complete, even under very strict hypothesis.

cs.DM↗

arXiv2020

Cycle convexity and the tunnel number of links

In this work, we introduce a new graph convexity, that we call Cycle Convexity, motivated by related notions in Knot Theory. For a graph $G=(V,E)$, define the interval function in the Cycle Convexity as $I_{cc}(S) = S\cup \{v\in V(G)\mid \text{there is a cycle }C\text{ in }G\text{ such that } V(C)\setminus S=\{v\}\}$, for every $S\subseteq V(G)$. We say that $S\subseteq V(G)$ is convex if $I_{cc}(S)=S$. The convex hull of $S\subseteq V(G)$, denoted by $Hull(S)$, is the inclusion-wise minimal convex set $S'$ such that $S\subseteq S'$. A set $S\subseteq V(G)$ is called a hull set if $Hull(S)=V(G)$. The hull number of $G$ in the cycle convexity, denoted by $hn_{cc}(G)$, is the cardinality of a smallest hull set of $G$. We first present the motivation for introducing such convexity and the study of its related hull number. Then, we prove that: the hull number of a 4-regular planar graph is at most half of its vertices; computing the hull number of a planar graph is an $NP$-complete problem; computing the hull humber of chordal graphs, $P_4$-sparse graphs and grids can be done in polynomial time.

math.CO↗

arXiv2020

On the proper orientation number of chordal graphs

An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is the number of arcs with head $v$ in $D$. An orientation $D$ of $G$ is proper if $d^-_D(u)\neq d^-_D(v)$, for all $uv\in E(G)$. An orientation with maximum indegree at most $k$ is called a $k$-orientation. The proper orientation number of $G$, denoted by $\overrightarrowχ(G)$, is the minimum integer $k$ such that $G$ admits a proper $k$-orientation. We prove that determining whether $\overrightarrowχ(G) \leq k$ is NP-complete for chordal graphs of bounded diameter, but can be solved in linear-time in the subclass of quasi-threshold graphs. When parameterizing by $k$, we argue that this problem is FPT for chordal graphs and argue that no polynomial kernel exists, unless $NP\subseteq coNP/\ poly$. We present a better kernel to the subclass of split graphs and a linear kernel to the class of cobipartite graphs. Concerning bounds, we prove tight upper bounds for subclasses of block graphs. We also present new families of trees having proper orientation number at most 2 and at most 3. Actually, we prove a general bound stating that any graph $G$ having no adjacent vertices of degree at least $c+1$ have proper orientation number at most $c$. This implies new classes of (outer)planar graphs with bounded proper orientation number. We also prove that maximal outerplanar graphs $G$ whose weak-dual is a path satisfy $\overrightarrowχ(G)\leq 13$. Finally, we present simple bounds to the classes of chordal claw-free graphs and cographs.

cs.CC↗

arXiv2020

Visualizing topological transport

We present a mathematically simple procedure for explaining and visualizing the dynamics of quantized transport in topological insulators. The procedure serves to illustrate and clarify the dynamics of topological transport in general, but for the sake of concreteness, it is phrased here in terms of electron transport in a charge-ordered chain, which may be mapped exactly onto transport between edge channels in the Integer Quantum Hall Effect. It has the advantage that it allows a direct visualization of the real-space and real-time evolution of the electronic charges throughout the topological pumping cycle, thus demystifying how charge flows between remote edges separated by an insulating bulk, why the amount of transported charge is given by a topological invariant, and how continuous driving yields a discrete, quantized amount of transported charge.

cond-mat.mes-hall↗

arXiv2020

A Unifying Model for Locally Constrained Spanning Tree Problems

Given a graph $G$ and a digraph $D$ whose vertices are the edges of $G$, we investigate the problem of finding a spanning tree of $G$ that satisfies the constraints imposed by $D$. The restrictions to add an edge in the tree depend on its neighborhood in $D$. Here, we generalize previously investigated problems by also considering as input functions $\ell$ and $u$ on $E(G)$ that give a lower and an upper bound, respectively, on the number of constraints that must be satisfied by each edge. The produced feasibility problem is denoted by \texttt{G-DCST}, while the optimization problem is denoted by \texttt{G-DCMST}. We show that \texttt{G-DCST} is NP-complete even under strong assumptions on the structures of $G$ and $D$, as well as on functions $\ell$ and $u$. On the positive side, we prove two polynomial results, one for \texttt{G-DCST} and another for \texttt{G-DCMST}, and also give a simple exponential-time algorithm along with a proof that it is asymptotically optimal under the Ð. Finally, we prove that other previously studied constrained spanning tree (\textsc{CST}) problems can be modeled within our framework, namely, the \textsc{Conflict CST}, the \textsc{Forcing CS, the \textsc{At Least One/All Dependency CST}, the \textsc{Maximum Degree CST}, the \textsc{Minimum Degree CST}, and the \textsc{Fixed-Leaves Minimum Degree CST}.

cs.CC↗

arXiv2020

Edge-Disjoint Branchings in Temporal Graphs

A temporal digraph ${\cal G}$ is a triple $(G, γ, λ)$ where $G$ is a digraph, $γ$ is a function on $V(G)$ that tells us the timestamps when a vertex is active, and $λ$ is a function on $E(G)$ that tells for each $uv \in E(G)$ when $u$ and $v$ are linked. Given a static digraph $G$, and a subset $R\subseteq V(G)$, a spanning branching with root $R$ is a subdigraph of $G$ that has exactly one path from $R$ to each $v\in V(G)$. In this paper, we consider the temporal version of Edmonds' classical result about the problem of finding $k$ edge-disjoint spanning branchings respectively rooted at given $R_1,\cdots,R_k$. We introduce and investigate different definitions of spanning branchings, and of edge-disjointness in the context of temporal graphs. A branching ${\cal B}$ is vertex-spanning if the root is able to reach each vertex $v$ of $G$ at some time where $v$ is active, while it is temporal-spanning if $v$ can be reached from the root at every time where $v$ is active. On the other hand, two branchings ${\cal B}_1$ and ${\cal B}_2$ are edge-disjoint if they do not use the same edge of $G$, and are temporal-edge-disjoint if they can use the same edge of $G$ but at different times. This lead us to four definitions of disjoint spanning branchings and we prove that, unlike the static case, only one of these can be computed in polynomial time, namely the temporal-edge-disjoint temporal-spanning branchings problem, while the other versions are $\mathsf{NP}$-complete, even under very strict assumptions.

cs.DS↗

arXiv2019

On Orthogonal Vector Edge Coloring

Given a graph $G$ and a positive integer $d$, an orthogonal vector $d$-coloring of $G$ is an assignment $f$ of vectors of $\mathbb{R}^d$ to $V(G)$ in such a way that adjacent vertices receive orthogonal vectors. The orthogonal chromatic number of $G$, denoted by $χ_v(G)$, is the minimum $d$ for which $G$ admits an orthogonal vector $d$-coloring. This notion has close ties with the notions of Lovász Theta Function, quantum chromatic number, and many other problems, and even though this and related metrics have been extensively studied over the years, we have found that there is a gap in the knowledge concerning the edge version of the problem. In this article, we discuss this version and its relation with other insteresting known facts, and pose a question about the orthogonal chromatic index of cubic graphs.

cs.DM↗

arXiv2019

b-continuity and Partial Grundy Coloring of graphs with large girth

A b-coloring of a graph is a proper coloring such that each color class has at least one vertex which is adjacent to each other color class. The b-spectrum of $G$ is the set $S_{b}(G)$ of integers $k$ such that $G$ has a b-coloring with $k$ colors and $b(G)=\max S_{b}(G)$ is the b-chromatic number of $G$. A graph is b-continous if $S_{b}(G)=[χ(G),b(G)]\cap \mathbb{Z}$. An infinite number of graphs that are not b-continuous is known. It is also known that graphs with girth at least 10 are b-continuous. A partial Grundy coloring is a proper coloring $f:V(G)\rightarrow \{1,\ldots,k\}$ such that each color class $i$ contains some vertex $u$ that is adjacent to every color class $j$ such that $j<i$. The partial Grundy number of $G$ is the maximum value $\partialΓ(G)$ for which $G$ has a partial Grundy coloring. In this work, we prove that graphs with girth at least 8 are b-continuous, and that the b-spectrum of a graph $G$ with girth at least 7 contains the integers between $2χ(G)$ and $b(G)$. We also prove that $\partialΓ(G)$ equals a known upper bound when $G$ is a graph with girth at least 7. These results generalize previous ones by Linhares-Sales and Silva (2017), and by Shi et al.(2005).

cs.DM↗

arXiv2018

Connected greedy coloring $H$-free graphs

A connected ordering $(v_1, v_2, \ldots, v_n)$ of $V(G)$ is an ordering of the vertices such that $v_i$ has at least one neighbour in $\{v_1, \ldots, v_{i - 1}\}$ for every $i \in \{2, \ldots, n\}$. A connected greedy coloring (CGC for short) is a coloring obtained by applying the greedy algorithm to a connected ordering. This has been first introduced in 1989 by Hertz and de Werra, but still very little is known about this problem. An interesting aspect is that, contrary to the traditional greedy coloring, it is not always true that a graph has a connected ordering that produces an optimal coloring; this motivates the definition of the connected chromatic number of $G$, which is the smallest value $χ_c(G)$ such that there exists a CGC of $G$ with $χ_c(G)$ colors. An even more interesting fact is that $χ_c(G) \le χ(G)+1$ for every graph $G$ (Benevides et. al. 2014). In this paper, in the light of the dichotomy for the coloring problem restricted to $H$-free graphs given by Král et.al. in 2001, we are interested in investigating the problems of, given an $H$-free graph $G$: (1). deciding whether $χ_c(G)=χ(G)$; and (2). given also a positive integer $k$, deciding whether $χ_c(G)\le k$. We have proved that Problem (2) has the same dichotomy as the coloring problem (i.e., it is polynomial when $H$ is an induced subgraph of $P_4$ or of $P_3+K_1$, and it is NP-complete otherwise). As for Problem (1), we have proved that $χ_c(G) = χ(G)$ always hold when $G$ is an induced subgraph of $P_5$ or of $P_4+K_1$, and that it is NP-hard to decide whether $χ_c(G)=χ(G)$ when $H$ is not a linear forest or contains an induced $P_9$. We mention that some of the results actually involve fixed $k$ and fixed $χ(G)$.

math.CO↗

arXiv2018

Dual parameterization of Weighted Coloring

Given a graph $G$, a proper $k$-coloring of $G$ is a partition $c = (S_i)_{i\in [1,k]}$ of $V(G)$ into $k$ stable sets $S_1,\ldots, S_{k}$. Given a weight function $w: V(G) \to \mathbb{R}^+$, the weight of a color $S_i$ is defined as $w(i) = \max_{v \in S_i} w(v)$ and the weight of a coloring $c$ as $w(c) = \sum_{i=1}^{k}w(i)$. Guan and Zhu [Inf. Process. Lett., 1997] defined the weighted chromatic number of a pair $(G,w)$, denoted by $σ(G,w)$, as the minimum weight of a proper coloring of $G$. The problem of determining $σ(G,w)$ has received considerable attention during the last years, and has been proved to be notoriously hard: for instance, it is NP-hard on split graphs, unsolvable on $n$-vertex trees in time $n^{o(\log n)}$ unless the ETH fails, and W[1]-hard on forests parameterized by the size of a largest tree. In this article we provide some positive results for the problem, by considering its so-called dual parameterization: given a vertex-weighted graph $(G,w)$ and an integer $k$, the question is whether $σ(G,w) \leq \sum_{v \in V(G)} w(v) - k$. We prove that this problem is FPT by providing an algorithm running in time $9^k \cdot n^{O(1)}$, and it is easy to see that no algorithm in time $2^{o(k)} \cdot n^{O(1)}$ exists under the ETH. On the other hand, we present a kernel with at most $(2^{k-1}+1) (k-1)$ vertices, and we rule out the existence of polynomial kernels unless ${\sf NP} \subseteq {\sf coNP} / {\sf poly}$, even on split graphs with only two different weights. Finally, we identify some classes of graphs on which the problem admits a polynomial kernel, in particular interval graphs and subclasses of split graphs, and in the latter case we present lower bounds on the degrees of the polynomials.

cs.DS↗

arXiv2018

The simple-cubic structure of elemental Polonium and its relation to combined charge and orbital order in other elemental chalcogens

Polonium is the only element to crystallise into a simple cubic structure at ambient conditions. Moreover, at high temperatures it undergoes a structural phase transition into a less symmetric trigonal configuration. It has long been suspected that the strong spin-orbit coupling in Polonium is involved in both peculiarities, but the precise mechanism by which it operates remains controversial. Here, we introduce a single microscopic model capable of capturing the atomic structure of all chalcogen crystals: Selenium, Tellurium, and Polonium. We show that the strong spin-orbit coupling in Polonium suppresses the trigonal charge and orbital ordered state known to be the ground state configuration of Selenium and Tellurium, and allows the simple cubic state to prevail instead. We also confirm a recent suggestion based on ab initio calculations that a small increase in the lattice constant may effectively decrease the role of spin-orbit coupling, leading to a re-emergence of the trigonal orbital ordered state at high temperatures. We conclude that Polonium is a unique element, in which spins, orbitals, electronic charges, and lattice deformations all cooperate and collectively cause the emergence of the only elemental crystal structure with the simplest possible, cubic, lattice.

cond-mat.str-el↗

arXiv2018

Weighted proper orientations of trees and graphs of bounded treewidth

Given a simple graph $G$, a weight function $w:E(G)\rightarrow \mathbb{N} \setminus \{0\}$, and an orientation $D$ of $G$, we define $μ^-(D) = \max_{v \in V(G)} w_D^-(v)$, where $w^-_D(v) = \sum_{u\in N_D^{-}(v)}w(uv)$. We say that $D$ is a weighted proper orientation of $G$ if $w^-_D(u) \neq w^-_D(v)$ whenever $u$ and $v$ are adjacent. We introduce the parameter weighted proper orientation number of $G$, denoted by $\overrightarrowχ(G,w)$, which is the minimum, over all weighted proper orientations $D$ of $G$, of $μ^-(D)$. When all the weights are equal to 1, this parameter is equal to the proper orientation number of $G$, which has been object of recent studies and whose determination is NP-hard in general, but polynomial-time solvable on trees. Here, we prove that the equivalent decision problem of the weighted proper orientation number (i.e., $\overrightarrowχ(G,w) \leq k$?) is (weakly) NP-complete on trees but can be solved by a pseudo-polynomial time algorithm whose running time depends on $k$. Furthermore, we present a dynamic programming algorithm to determine whether a general graph $G$ on $n$ vertices and treewidth at most ${\sf tw}$ satisfies $\overrightarrowχ(G,w) \leq k$, running in time $O(2^{{\sf tw}^2}\cdot k^{3{\sf tw}}\cdot {\sf tw} \cdot n)$, and we complement this result by showing that the problem is W[1]-hard on general graphs parameterized by the treewidth of $G$, even if the weights are polynomial in $n$.

cs.DS↗

arXiv2017

On the complexity of finding internally vertex-disjoint long directed paths

For two positive integers $k$ and $\ell$, a $(k \times \ell)$-spindle is the union of $k$ pairwise internally vertex-disjoint directed paths with $\ell$ arcs between two vertices $u$ and $v$. We are interested in the (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle, which generalize both the Maximum Flow and Longest Path problems. We obtain the following complexity dichotomy: for a fixed $\ell \geq 1$, finding the largest $k$ such that an input digraph $G$ contains a subdivision of a $(k \times \ell)$-spindle is polynomial-time solvable if $\ell \leq 3$, and NP-hard otherwise. We place special emphasis on finding spindles with exactly two paths and present FPT algorithms that are asymptotically optimal under the ETH. These algorithms are based on the technique of representative families in matroids, and use also color-coding as a subroutine. Finally, we study the case where the input graph is acyclic, and present several algorithmic and hardness results.

cs.DS↗

arXiv2017

Elemental chalcogens as a minimal model for chiral charge and orbital order

Helices of increased electron density can spontaneously form in materials containing multiple, interacting density waves. Although a macroscopic order parameter theory describing this behaviour has been proposed and experimentally tested, a detailed microscopic understanding of spiral electronic order in any particular material is still lacking. Here, we present the elemental chalcogens Selenium and Tellurium as model materials for the development of chiral charge and orbital order. We formulate minimal models capturing the formation of spiral structures both in terms of a macroscopic Landau theory and a microscopic Hamiltonian. Both reproduce the known chiral crystal structure and are consistent with its observed thermal evolution and behaviour under applied pressure. The combination of microscopic and macroscopic frameworks allows us to distil the essential ingredients in the emergence of helical charge order, and may serve as a guide to understanding spontaneous chirality both in other specific materials and throughout materials classes.

cond-mat.str-el↗

arXiv2016

On the b-continuity of the lexicographic product of graphs

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of $G$ is the maximum integer $χ_b(G)$ for which $G$ has a b-coloring with $χ_b(G)$ colors. A graph $G$ is b-continuous if $G$ has a b-coloring with $k$ colors, for every integer $k$ in the interval $[χ(G),χ_b(G)]$. It is known that not all graphs are b-continuous. Here, we investigate whether the lexicographic product $G[H]$ of b-continuous graphs $G$ and $H$ is also b-continuous. Using homomorphisms, we provide a new lower bound for $χ_b(G[H])$, namely $χ_b(G[K_t])$, where $t=χ_b(H)$, and prove that if $G[K_\ell]$ is b-continuous for every positive integer $\ell$, then $G[H]$ admits a b-coloring with $k$ colors, for every $k$ in the interval $[χ(G[H]),χ_b(G[K_t])]$. We also prove that $G[K_\ell]$ is b-continuous, for every positive integer $\ell$, whenever $G$ is a $P_4$-sparse graph, and we give further results on the b-spectrum of $G[K_\ell]$, when $G$ is chordal.

cs.DM↗

arXiv2016

Fall-colorings and b-colorings of graph products

Given a proper coloring $f$ of a graph $G$, a b-vertex in $f$ is a vertex that is adjacent to every color class but its own. It is a b-coloring if every color class contains at least one b-vertex, and it is a fall-coloring if every vertex is a b-vertex. The b-chromatic number of $G$ is the maximum integer $b(G)$ for which $G$ has a b-coloring with $b(G)$ colors, while the fall-chromatic number and the fall-acromatic number of $G$ are, respectively, the minimum and maximum integers $f_1(G),f_2(G)$ for which $G$ has a fall-coloring. In this article, we explore the concepts of b-homomorphisms and Type II homomorphisms, which generalize the concepts of b-colorings and fall-colorings, and present some meta-theorems concerning products of graphs. As a result, we derive some previously known facts about these metrics on graph products. We also give a negative answer to a question posed by Kaul and Mitillos about fall-colorings of perfect graphs.

math.CO↗

arXiv2016

The b-continuity of graphs with large girth

math.CO↗

arXiv2016

Circular Backbone Colorings: on matching and tree backbones of planar graphs

Given a graph $G$, and a spanning subgraph $H$ of $G$, a circular $q$-backbone $k$-coloring of $(G,H)$ is a proper $k$-coloring $c$ of $G$ such that $q\le \lvert c(u)-c(v)\rvert \le k-q$, for every edge $uv\in E(H)$. The circular $q$-backbone chromatic number of $(G,H)$, denoted by $CBC_q(G,H)$, is the minimum integer $k$ for which there exists a circular $q$-backbone $k$-coloring of $(G,H)$. The Four Color Theorem implies that whenever $G$ is planar, we have $CBC_2(G,H)\le 8$. It is conjectured that this upper bound can be improved to 7 when $H$ is a tree, and to 6 when $H$ is a matching. In this work, we show that: 1) if $G$ is planar and has no $C_4$ as subgraph, and $H$ is a linear spanning forest of $G$, then $CBC_2(G,H)\leq 7$; 2) if $G$ is a plane graph having no two 3-faces sharing an edge, and $H$ is a matching of $G$, then $CBC_2(G,H)\leq 6$; and 3) if $G$ is planar and has no $C_4$ nor $C_5$ as subgraph, and $H$ is a mathing of $G$, then $CBC_2(G,H)\leq 5$. These results partially answer questions posed by Broersma, Fujisawa and Yoshimoto (2003), and by Broersma, Fomin and Golovach (2007). It also points towards a positive answer for the Steinberg's Conjecture.

cs.DM↗