arXiv subjects

Thomas Bellitto

Publications and source records attributed to Thomas Bellitto.

At least 19 recordsLinked to original sources

arXiv2026

Designing sparse temporal graphs satisfying connectivity requirements

Connectivity of temporal graphs has been widely studied both as graph theory and as gossip theory. In particular, it is well known that in order to connect every vertex to every other, a temporal graph needs to have at least $2n-4$ edges where $n$ is the number of vertices. This paper investigates the optimal number of edges required to satisfy partial connectivity requirements. We introduce the problem of Connectivity Request Satisfaction where we are given a directed graph that we call the request graph, where an arc from $u$ to $v$ means that we need to be able to go from $u$ to $v$. Our goal is to build a temporal graph on the same vertex set with as few temporal edges as possible that would satisfy all the requests. When the graph we build is directed, we prove that the number of temporal arcs required is $n-\mathrm{cc}+\mathrm{dfvs}$ where $\mathrm{cc}$ is the number of connected component of the request graph and $\mathrm{dfvs}$ is the size of its smallest directed feedback vertex set. It follows that the problem is NP-complete but inherits fixed parameter tractability properties of Directed Feedback Vertex Set. When the graph we build is undirected, we establish a characterization of strongly connected request graphs that admit a solution with $n-1$ edges: it is possible if and only if any set of pairwise non-vertex-disjoint closed walks all share a common vertex. We prove that this criteria can be tested in polynomial time.

cs.DS↗

arXiv2026

Inversion diameter and 2-edge-colored homomorphisms

In an oriented graph, the inversion of a subset of vertices X is the operation reversing the direction of every arc with both endpoints in X. Given a graph G, the inversion distance between two orientations G is the minimum number of inversions transforming one into the other. The inversion diameter diam(G) is the maximum such distance over all pairs of orientations of G. Through an equivalent formulation of inversions over 2-edge-colorings of G, we introduce the use of homomorphism-universal 2-edge-colored graphs to obtain bounds on the inversion diameter of various classes of graphs. Our first result upper bounds the inversion diameter by a linear function of the acyclic chromatic number, improving on the previous quadratic dependency. We then consider the inversion diameter of planar graphs, exhibiting a lower bound of 6, as well as new lower and upper bounds for those of a given girth, in particular settling the girth 7 case. We then show that any triangle-free graph G with maximum degree D satisfies diam(G) <= D + log D, making progress on the conjecture of Havet et al. that diam(G) <= D. Finally, we prove a general result about subdivisions: if a graph has inversion diameter k, any of its subdivisions has inversion diameter at most k + log k + 5.

math.CO↗

arXiv2024

Canadian Traveller Problems in Temporal Graphs

This paper formalises the Canadian Traveller problem as a positional two-player game on graphs. We consider two variants depending on whether an edge is blocked. In the locally-informed variant, the traveller learns if an edge is blocked upon reaching one of its endpoints, while in the uninformed variant, they discover this only when the edge is supposed to appear. We provide a polynomial algorithm for each shortest path variant in the uninformed case. This algorithm also solves the case of directed acyclic non-temporal graphs. In the locally-informed case, we prove that finding a winning strategy is PSPACE-complete. Moreover, we establish that the problem is polynomial-time solvable when $k=1$ but NP-hard for $k\geq 2$. Additionally, we show that the standard (non-temporal) Canadian Traveller Problem is NP-hard when there are $k\geq 4$ blocked edges, which is, to the best of our knowledge, the first hardness result for CTP for a constant number of blocked edges.

cs.DS↗

arXiv2022

The smallest 5-chromatic tournament

A coloring of a digraph is a partition of its vertex set such that each class induces a digraph with no directed cycles. A digraph is $k$-chromatic if $k$ is the minimum number of classes in such partition, and a digraph is oriented if there is at most one arc between each pair of vertices. Clearly, the smallest $k$-chromatic digraph is the complete digraph on $k$ vertices, but determining the order of the smallest $k$-chromatic oriented graphs is a challenging problem. It is known that the smallest $2$-, $3$- and $4$-chromatic oriented graphs have $3$, $7$ and $11$ vertices, respectively. In 1994, Neumann-Lara conjectured that a smallest $5$-chromatic oriented graph has $17$ vertices. We solve this conjecture and show that the correct order is $19$.

cs.DM↗

arXiv2022

Homomorphically Full Oriented Graphs

Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete.

cs.DM↗

arXiv2022

On the minimum number of arcs in $k$-dicritical oriented graphs

The dichromatic number $\dic(D)$ of a digraph $D$ is the least integer $k$ such that $D$ can be partitioned into $k$ directed acyclic digraphs. A digraph is $k$-dicritical if $\dic(D) = k$ and each proper subgraph $D'$ of $D$ satisfies $\dic(D') \leq k-1$. An oriented graph is a digraph with no directed cycle of length $2$. For integers $k$ and $n$, we denote by $o_k(n)$ the minimum number of edges of a $k$-critical oriented graph on $n$ vertices (with the convention $o_k(n)=+\infty$ if there is no $k$-dicritical oriented graph of order $n$). The main result of this paper is a proof that $o_3(n) \geq \frac{7n+2}{3}$ together with a construction witnessing that $o_3(n) \leq \left \lceil \frac{5n}{2} \right \rceil$ for all $n \geq 12$. We also give a construction showing that for all sufficiently large $n$ and all $k\geq 3$, $o_k(n) < (2k-3)n$, disproving a conjecture of Hoshino and Kawarabayashi. Finally, we prove that, for all $k\geq 2$, $o_k(n) \geq \pth{ k - \frac{3}{4}-\frac{1}{4k-6}} n + \frac{3}{4(2k-3)}$, improving the previous best known lower bound of Bang-Jensen, Bellitto, Schweser and Stiebitz.

math.CO↗

arXiv2021

Locating Dominating Sets in local tournaments

A dominating set in a directed graph is a set of vertices $S$ such that all the vertices that do not belong to $S$ have an in-neighbour in $S$. A locating set $S$ is a set of vertices such that all the vertices that do not belong to $S$ are characterized uniquely by the in-neighbours they have in $S$, i.e. for every two vertices $u$ and $v$ that are not in $S$, there exists a vertex $s\in S$ that dominates exactly one of them. The size of a smallest set of a directed graph $D$ which is both locating and dominating is denoted by $\gamma^{LD}(D)$. Foucaud, Heydarshahi and Parreau proved that any twin-free digraph $D$ satisfies $\gamma^{LD}(D)\leq \frac{4n} 5 +1$ but conjectured that this bound can be lowered to $\frac{2n} 3$. The conjecture is still open. They also proved that if $D$ is a tournament, i.e. a directed graph where there is one arc between every pair of vertices, then $\gamma^{LD}(D)\leq \lceil \frac{n}{2}\rceil$. The main result of this paper is the generalization of this bound to connected local tournaments, i.e. connected digraphs where the in- and out-neighbourhoods of every vertex induce a tournament. We also prove $\gamma^{LD}(D)\leq \frac{2n} 3$ for all quasi-twin-free digraphs $D$ that admit a supervising vertex (a vertex from which any vertex is reachable). This class of digraphs generalizes twin-free acyclic graphs, the most general class for which this bound was known.

cs.DM↗

arXiv2021

Close relatives (of Feedback Vertex Set), revisited

At IPEC 2020, Bergougnoux, Bonnet, Brettell, and Kwon showed that a number of problems related to the classic Feedback Vertex Set (FVS) problem do not admit a $2^{o(k \log k)} \cdot n^{\mathcal{O}(1)}$-time algorithm on graphs of treewidth at most $k$, assuming the Exponential Time Hypothesis. This contrasts with the $3^{k} \cdot k^{\mathcal{O}(1)} \cdot n$-time algorithm for FVS using the Cut&Count technique. During their live talk at IPEC 2020, Bergougnoux et al.~posed a number of open questions, which we answer in this work. - Subset Even Cycle Transversal, Subset Odd Cycle Transversal, Subset Feedback Vertex Set can be solved in time $2^{\mathcal{O}(k \log k)} \cdot n$ in graphs of treewidth at most $k$. This matches a lower bound for Even Cycle Transversal of Bergougnoux et al.~and improves the polynomial factor in some of their upper bounds. - Subset Feedback Vertex Set and Node Multiway Cut can be solved in time $2^{\mathcal{O}(k \log k)} \cdot n$, if the input graph is given as a clique-width expression of size $n$ and width $k$. - Odd Cycle Transversal can be solved in time $4^k \cdot k^{\mathcal{O}(1)} \cdot n$ if the input graph is given as a clique-width expression of size $n$ and width $k$. Furthermore, the existence of a constant $\varepsilon > 0$ and an algorithm performing this task in time $(4-\varepsilon)^k \cdot n^{\mathcal{O}(1)}$ would contradict the Strong Exponential Time Hypothesis.

cs.DM↗

arXiv2020

The Complexity of Connectivity Problems in Forbidden-Transition Graphs and Edge-Colored Graphs

The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs. In a forbidden-transition graph, every pair of edges incident to a common vertex is permitted or forbidden; a walk is compatible if all pairs of consecutive edges on the walk are permitted. Forbidden-transition graphs and related models have found applications in a variety of fields, such as routing in optical telecommunication networks, road networks, and bio-informatics. We initiate the study of fundamental connectivity problems from the point of view of parameterized complexity, including an in-depth study of tractability with regards to various graph-width parameters. Among several results, we prove that finding a simple compatible path between given endpoints in a forbidden-transition graph is W[1]-hard when parameterized by the vertex-deletion distance to a linear forest (so it is also hard when parameterized by pathwidth or treewidth). On the other hand, we show an algebraic trick that yields tractability when parameterized by treewidth of finding a properly colored Hamiltonian cycle in an edge-colored graph; properly colored walks in edge-colored graphs is one of the most studied special cases of compatible walks in forbidden-transition graphs.

cs.DS↗

arXiv2020

Supereulerian 2-edge-coloured graphs

A 2-edge-coloured graph $G$ is {\bf supereulerian} if $G$ contains a spanning closed trail in which the edges alternate in colours. An {\bf eulerian factor} of a 2-edge-coloured graph is a collection of vertex disjoint induced subgraphs which cover all the vertices of $G$ such that each of these subgraphs is supereulerian. We give a polynomial algorithm to test if a 2-edge-coloured graph has an eulerian factor and to produce one when it exists. A 2-edge-coloured graph is {\bf (trail-)colour-connected} if it contains a pair of alternating $(u,v)$-paths ($(u,v)$-trails) whose union is an alternating closed walk for every pair of distinct vertices $u,v$. A 2-edge-coloured graph is {\bf M-closed} if $xz$ is an edge of $G$ whenever some vertex $u$ is joined to both $x$ and $z$ by edges of the same colour. M-closed 2-edge-coloured graphs, introduced in \cite{balbuenaDMTCS21}, form a rich generalization of 2-edge-coloured complete graphs. We show that if $G$ is an extension of an M-closed 2-edge-coloured complete graph, then $G$ is supereulerian if and only if $G$ is trail-colour-connected and has an eulerian factor. We also show that for general 2-edge-coloured graphs it is NP-complete to decide whether the graph is supereulerian. Finally we pose a number of open problems.

math.CO↗

arXiv2019

Counterexamples to Thomassen's conjecture on decomposition of cubic graphs

We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.

math.CO↗

arXiv2019

Haj\'os and Ore constructions for digraphs

The chromatic number $\overrightarrow{\chi}(D)$ of a digraph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class induces an acyclic subdigraph of $D$. A digraph $D$ is $k$-critical if $\overrightarrow{\chi}(D) = k$ but $\overrightarrow{\chi}(D') < k$ for all proper subdigraphs $D'$ of $D$. We examine methods for creating infinite families of critical digraphs, the Dirac join and the directed and bidirected Haj\'os join. We prove that a digraph $D$ has chromatic number at least $k$ if and only if it contains a subdigraph that can be obtained from bidirected complete graphs on $k$ vertices by (directed) Haj\'os joins and identifying non-adjacent vertices. Building upon that, we show that a digraph $D$ has chromatic number at least $k$ if and only if it can be constructed from bidirected $K_k$'s by using directed and bidirected Haj\'os joins and identifying non-adjacent vertices (so called Ore joins), thereby transferring a well-known result of Urquhart to digraphs. Finally, we prove a Gallai-type theorem that characterizes the structure of the low vertex subdigraph of a critical digraph, that is, the subdigraph, which is induced by the vertices that have in-degree $k-1$ and out-degree $k-1$ in $D$.

math.CO↗

arXiv2019

The directed 2-linkage problem with length constraints

The {\sc weak 2-linkage} problem for digraphs asks for a given digraph and vertices $s_1,s_2,t_1,t_2$ whether $D$ contains a pair of arc-disjoint paths $P_1,P_2$ such that $P_i$ is an $(s_i,t_i)$-path. This problem is NP-complete for general digraphs but polynomially solvable for acyclic digraphs \cite{fortuneTCS10}. Recently it was shown \cite{bercziESA17} that if $D$ is equipped with a weight function $w$ on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak-2-linkage problem when both paths have to be shortest paths in $D$. In this paper we consider the unit weight case and prove that for every pair constants $k_1,k_2$, there is a polynomial algorithm which decides whether the input digraph $D$ has a pair of arc-disjoint paths $P_1,P_2$ such that $P_i$ is an $(s_i,t_i)$-path and the length of $P_i$ is no more than $d(s_i,t_i)+k_i$, for $i=1,2$, where $d(s_i,t_i)$ denotes the length of the shortest $(s_i,t_i)$-path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution $P_1,P_2$ to the {\sc weak 2-linkage} problem where each path $P_i$ has length at most $d(s_i,t_i)+ c\log^{1+\epsilon}{}n$ for some constant $c$. We also prove that the {\sc weak 2-linkage} problem remains NP-complete if we require one of the two paths to be a shortest path while the other path has no restriction on the length.

cs.CC↗

arXiv2019

Proper-walk connection number of graphs

This paper studies the problem of proper-walk connection number: given an undirected connected graph, our aim is to colour its edges with as few colours as possible so that there exists a properly coloured walk between every pair of vertices of the graph i.e. a walk that does not use consecutively two edges of the same colour. The problem was already solved on several classes of graphs but still open in the general case. We establish that the problem can always be solved in polynomial time in the size of the graph and we provide a characterization of the graphs that can be properly connected with $k$ colours for every possible value of $k$.

cs.DM↗

arXiv2018

On DP-Coloring of Digraphs

DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph $G$ with a list-assignment $L$ to finding an independent transversal in an auxiliary graph with vertex set $\{(v,c) ~|~ v \in V(G), c \in L(v)\}$. In this paper, we extend the definition of DP-colorings to digraphs using the approach from Neumann-Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number, which extends various results on the (list-)chromatic number of digraphs.

math.CO↗

arXiv2018

On the density of sets of the Euclidean plane avoiding distance 1

A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall x,y \in A, \left\| x-y \right\|_2 \neq 1.$ In this paper we study the number $m_1(\mathbb R^2)$ which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, $m_1(\mathbb R^2)$ represents the highest proportion of the plane that can be filled by a set avoiding distance 1. This parameter is related to the fractional chromatic number $\chi_f(\mathbb R^2)$ of the plane. We establish that $m_1(\mathbb R^2) \leq 0.25647$ and $\chi_f(\mathbb R^2) \geq 3.8991$.

math.MG↗

arXiv2018

On Minimum Connecting Transition Sets in Graphs

A forbidden transition graph is a graph defined together with a set of permitted transitions i.e. unordered pair of adjacent edges that one may use consecutively in a walk in the graph. In this paper, we look for the smallest set of transitions needed to be able to go from any vertex of the given graph to any other. We prove that this problem is NP-hard and study approximation algorithms. We develop theoretical tools that help to study this problem.

cs.DS↗

arXiv2017

Complexity of locally-injective homomorphisms to tournaments

For oriented graphs $G$ and $H$, a homomorphism $f: G \rightarrow H$ is locally-injective if, for every $v \in V(G)$, it is injective when restricted to some combination of the in-neighbourhood and out-neighbourhood of $v$. Two of the possible definitions of local-injectivity are examined. In each case it is shown that the associated homomorphism problem is NP-complete when $H$ is a reflexive tournament on three or more vertices with a loop at every vertex, and solvable in polynomial time when $H$ is a reflexive tournament on two or fewer vertices.

cs.DM↗