arXiv subjects

Allen Ibiapina

Publications and source records attributed to Allen Ibiapina.

10 recordsLinked to original sources

arXiv2026

Complexity Gaps between Point and Interval Temporal Graphs for some Reachability Problems

Temporal graphs arise when modeling interactions that evolve over time. They usually come in several flavors, depending on the number of parameters used to describe the temporal aspects of the interactions: time of appearance, duration, delay of transmission. In the point model, edges appear at specific points in time, whereas in the more general interval model, edges can be present over specific time intervals. In both models, the delay for traversing an edge can change with each edge appearance. When time is discrete, the two models are equivalent in the sense that the presence of an edge during an interval is equivalent to a sequence of point-in-time occurrences of the edge. However, this transformation can drastically change the size of the input and has implications for complexity. Indeed, we show a gap between the two models with respect to the complexity of the classical problem of computing a fastest temporal path from a source vertex to a target vertex, i.e. a path where edges can be traversed one after another in time and such that the total duration from source to target is minimized. It can be solved in near-linear time in the point model, while we show that the interval model requires quadratic time under classical assumptions of fine-grained complexity. With respect to linear time, our lower bound implies a factor of the number of vertices, while the best known algorithm has a factor of the number of underlying edges. We also show a similar complexity gap for computing a shortest temporal path, i.e. a temporal path with a minimum number of edges. Here our lower bound matches known upper bounds up to a logarithmic factor. Interestingly, we show that near-linear time for fastest temporal path computation is possible in the interval model when it is restricted to uniform delay zero, i.e., when traversing an edge is instantaneous. However, this special case is not exempt from our lower bound for shortest temporal path computation. These two results should be contrasted with the computation of a foremost temporal path, i.e., a temporal path that arrives as early as possible. It is well known that this computation can be solved in near-linear time in both models. We also show that there is no gap in testing the all-to-all temporal connectivity of a temporal graph. We demonstrate a quadratic lower bound that applies to both the interval and point models and aligns with the existing upper bounds.

cs.DS↗

arXiv2025

Extension of the Gyárfás-Sumner conjecture to signed graphs

The balanced chromatic number of a signed graph G is the minimum number of balanced sets that cover all vertices of G. Studying structural conditions which imply bounds on the balanced chromatic number of signed graphs is among the most fundamental problems in graph theory. In this work, we initiate the study of coloring hereditary classes of signed graphs. More precisely, we say that a set F = {F_1, F_2, ..., F_l} is a GS (for Gyárfás-Sumner) set if there exists a constant c such that signed graphs with no induced subgraph switching equivalent to a member of F admit a balanced c-coloring. The focus of this work is to study GS sets of order 2. We show that if F is a GS set of order 2, then F_1 is either (K_3, -) or (K_4, -), and F_2 is a linear forest. In the case of F_1 = (K_3, -), we show that any choice of a linear forest for F_2 works. In the case of F_1 = (K_4, -), we show that if each connected component of F_2 is a path of length at most 4, then {F_1, F_2} is a GS set.

math.CO↗

arXiv2025

Menger's Theorem for Temporal Paths (Not Walks)

A (directed) temporal graph is a (directed) graph whose edges are available only at specific times during its (discretized) lifetime $τ$. In this setting, we ask that walks respect the temporal aspect by defining $\textit{temporal walks}$ as sequences of adjacent edges whose appearing times are either strictly increasing or non-decreasing (here called non-strict), depending on the scenario. The notion of disjointness between walks is also not unique: two walks are $\textit{vertex-disjoint}$ if they do not share a vertex, and are $\textit{temporal vertex-disjoint}$ if they do not share a vertex at the same time. Thus a $\textit{temporal path}$ is a temporal walk where no repetition of vertices, at any time, is allowed. This is an important distinction that separates the interpretation of our results from those of previous works on the topic. In this paper we focus on various questions regarding connectivity (maximum number of disjoint paths) and robustness (minimum size of a cut) between a given pair of vertices. Such problems are related to the well-known Menger's Theorem on static graphs. We explore all possible interpretations of such problems, according to vertex and temporal vertex-disjointness, strict and non-strict temporal paths, and directed and undirected temporal graphs. We present a number of new results, the main of which states that Menger's Theorem holds when the maximum number of temporal vertex-disjoint temporal paths is equal to 1.

cs.DM↗

arXiv2025

How to Color Temporal Graphs to Ensure Proper Transitions

Graph Coloring consists in assigning colors to vertices ensuring that two adjacent vertices do not have the same color. In dynamic graphs, this notion is not well defined, as we need to decide if different colors for adjacent vertices must happen all the time or not, and how to go from a coloring in one time to the next one. In this paper, we define a coloring notion for Temporal Graphs where at each step, the coloring must be proper. It uses a notion of compatibility between two consecutive snapshots that implies that the coloring stays proper while the transition happens. Given a graph, the minimum number of colors needed to ensure that such coloring exists is the \emph{Temporal Chromatic Number} of this graph. With those notions, we provide some lower and upper bounds for the temporal chromatic number in the general case. We then dive into some specific classes of graphs such as trees, graphs with bounded degree or bounded degeneracy. Finally, we consider temporal graphs where grow pace is one, that is, a single edge can be added and a single other one can be removed between two time steps. In that case, we consider bipartite and bounded degree graphs. Even though the problem is defined with full knowledge of the temporal graph, our results also work in the case where future snapshots are given online: we need to choose the coloring of the next snapshot after having computed the current one, not knowing what

cs.DM↗

arXiv2025

Temporal Cycle Detection and Acyclic Temporization

In directed graphs, a cycle can be seen as a structure that allows its vertices to loop back to themselves, or as a structure that allows pairs of vertices to reach each other through distinct paths. We extend these concepts to temporal graph theory, resulting in multiple interesting definitions of a "temporal cycle". For each of these, we consider the problems of Cycle Detection and Acyclic Temporization. For the former, we are given an input temporal digraph, and we want to decide whether it contains a temporal cycle. Regarding the latter, for a given input (static) digraph, we want to time the arcs such that no temporal cycle exists in the resulting temporal digraph. We're also interested in Acyclic Temporization where we bound the lifetime of the resulting temporal digraph. Multiple results are presented, including polynomial and fixed-parameter tractable search algorithms, polynomial-time reductions from 3-SAT and Not All Equal 3-SAT, and temporizations resulting from arbitrary vertex orderings which cover (almost) all cases.

cs.CC↗

arXiv2023

Snapshot disjointness in temporal graphs

In the study of temporal graphs, only paths respecting the flow of time are relevant. In this context, many concepts of walks disjointness were proposed over the years, and the validity of Menger's Theorem, as well as the complexity of related problems, has been investigated. In this paper, we introduce and investigate a type of disjointness that is only time dependent. Two walks are said to be snapshot disjoint if they are not active in a same snapshot (also called timestep). The related paths and cut problems are then defined and proved to be W[1]-hard and XP-time solvable when parameterized by the size of the solution. Additionally, in the light of the definition of Mengerian graphs given by Kempe, Kleinberg and Kumar in their seminal paper (STOC'2000), we define a Mengerian graph for time as a graph $G$ that cannot form an example where Menger's Theorem does not hold in the context of snapshot disjointness. We then give a characterization in terms of forbidden structures and provide a polynomial-time recognition algorithm. Finally, we also prove that, given a temporal graph $(G,λ)$ and a pair of vertices $s,z\in V(G)$, deciding whether at most $h$ multiedges can separate $s$ from $z$ is NP-complete.

cs.DM↗

arXiv2022

Mengerian graphs: characterization and recognition

A temporal graph ${\cal G}$ is a graph that changes with time. More specifically, it is a pair $(G, λ)$ where $G$ is a graph and $λ$ is a function on the edges of $G$ that describes when each edge $e\in E(G)$ is active. Given vertices $s,t\in V(G)$, a temporal $s,t$-path is a path in $G$ that traverses edges in non-decreasing time; and if $s,t$ are non-adjacent, then a temporal $s,t$-cut is a subset $S\subseteq V(G)\setminus\{s,t\}$ whose removal destroys all temporal $s,t$-paths. It is known that Menger's Theorem does not hold on this context, i.e., that the maximum number of internally vertex disjoint temporal $s,t$-paths is not necessarily equal to the minimum size of a temporal $s,t$-cut. In a seminal paper, Kempe, Kleinberg and Kumar (STOC'2000) defined a graph $G$ to be Mengerian if equality holds on $(G,λ)$ for every function $λ$. They then proved that, if each edge is allowed to be active only once in $(G,λ)$, then $G$ is Mengerian if and only if $G$ has no gem as topological minor. In this paper, we generalize their result by allowing edges to be active more than once, giving a characterization also in terms of forbidden structures. We additionally provide a polynomial time recognition algorithm.

math.CO↗

arXiv2021

Coloring Problems on Bipartite Graphs of Small Diameter

We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we investigate the $k$-List Coloring, List $k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with diameter at most $d$, proving NP-completeness in most cases, and leaving open only the List $3$-Coloring and $3$-Precoloring Extension problems when $d=3$. Some of these results are obtained through a proof that the Surjective $C_6$-Homomorphism problem is NP-complete on bipartite graphs with diameter at most four. Although the latter result has been already proved [Vikas, 2017], we present ours as an alternative simpler one. As a byproduct, we also get that $3$-Biclique Partition is NP-complete. An attempt to prove this result was presented in [Fleischner, Mujuni, Paulusma, and Szeider, 2009], but there was a flaw in their proof, which we identify and discuss here. Finally, we prove that the $3$-Fall Coloring problem is NP-complete on bipartite graphs with diameter at most four, and prove that NP-completeness for diameter three would also imply NP-completeness of $3$-Precoloring Extension on diameter three, thus closing the previously mentioned open cases. This would also answer a question posed in [Kratochvíl, Tuza, and Voigt, 2002].

math.CO↗

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↗