arXiv subjects

Cláudia Linhares Sales

Publications and source records attributed to Cláudia Linhares Sales.

7 recordsLinked to original sources

arXiv2026

Star Coloring on Some Subclasses of Chordal Graphs

A star coloring of a graph $G$ is a proper coloring in which no path on four vertices is bicolored. The star chromatic number $χ_{\star}(G)$ is the minimum number of colors in a star coloring of $G$. In this work we study star colorings from the perspective of forbidden induced subgraphs, focusing on three subclasses of chordal graphs. We provide both a structural characterization and a characterization in terms of forbidden induced subgraphs for star $3$-colorable chordal graphs; such characterizations yield a simple certifying recognition algorithm, running in time $O(|V|+|E|)$, for this class. We also characterize split graphs that are star $4$-colorable and star $5$-colorable in terms of (finitely many) forbidden induced subgraphs, again deriving linear-time certifying recognition algorithms. Finally, we study star colorings of $2$-trees and $2$-paths: we characterize the $2$-paths that are star $4$-colorable, prove that every $2$-path is star $5$-colorable, and exhibit a $2$-tree on $21$ vertices with star chromatic number $6$ such that any proper induced subgrahp has star chromatic number $5$.

math.CO↗

arXiv2026

Maya-Tupi graphs: a generalization of split graphs

We define the family of Maya-Tupi graphs as those graphs that admit a partition $(A,B)$ of their vertex sets such that $A$ induces a complete multipartite graph where each part has size at most two, and $B$ induces a graph where every connected component is $K_1$ or $K_2$. The family of Maya-Tupi graphs is self complementary, generalizes split graphs, falls into the sparse-dense partitioning schema and is characterized by finitely many forbidden induced subgraphs. Unfortunately, our computational experiments show that the number of minimal forbidden induced subgraphs to characterize Maya-Tupi graphs is greater than 2000. In this work, we find characterizations in terms of minimal forbidden induced subgraphs for disconnected graphs, which imply the same for cographs; our results imply linear-time certifying recognition algorithms for Maya-Tupi graphs within these classes. We also show that Maya-Tupi graphs can be recognized in $\mathcal{O}(n^3)$-time in $C_4$-free graphs and in graphs with bounded neighborhood diversity; in $\mathcal{O}(n^4)$-time for triangle-free graphs; and in $\mathcal{O}(n^2)$-time for graphs with bounded clique-width. We provide efficient algorithms to calculate the clique, the independence, the chromatic, and the treewidth numbers, as well as a minimum fill-in for Maya-Tupi graphs.

math.CO↗

arXiv2025

Minimum cost flow decomposition on arc-coloured networks

A network $\mathcal{N}$ is formed by a (multi)digraph $D$ together with a \emph{capacity function} $u : A(D) \to R_+$, and it is denoted by $\mathcal{N} = (D,u)$. A flow on $\mathcal{N}$ is a function $x: A(D) \to R_+$ such that $x(a) \leq u(a)$ for all $a \in A(D)$, and it is said to be $k$-splittable if it can be decomposed into up to $k$ paths. We say that a flow is $λ$-uniform if its value on each arc of the network with positive flow value is exactly $λ$, for some $λ\in R_+^*$. Arc-coloured networks are used to model qualitative differences among different regions through which the flow will be sent. They have applications in several areas such as communication networks, multimodal transportation, molecular biology, packing etc. We consider the problem of decomposing a flow over an arc-coloured network with minimum cost, that is, with minimum sum of the cost of its paths, where the cost of each path is given by its number of colours. We show that this problem is NP-Hard for general flows. When we restrict the problem to $λ$-uniform flows, we show that it can be solved in polynomial time for networks with at most two colours, and it is NP-Hard for general networks with three colours and for acyclic networks with at least five colours.

cs.CC↗

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

Minimal obstructions to $2$-polar cographs

A graph is a cograph if it is $P_4$-free. A $k$-polar partition of a graph $G$ is a partition of the set of vertices of $G$ into parts $A$ and $B$ such that the subgraph induced by $A$ is a complete multipartite graph with at most $k$ parts, and the subgraph induced by $B$ is a disjoint union of at most $k$ cliques with no other edges. It is known that $k$-polar cographs can be characterized by a finite family of forbidden induced subgraphs, for any fixed $k$. A concrete family of such forbidden induced subgraphs is known for $k=1$, since $1$-polar graphs are precisely split graphs. For larger $k$ such families are not known, and Ekim, Mahadev, and de Werra explicitely asked for the family for $k=2$. In this paper we provide such a family, and show that the graphs can be obtained from four basic graphs by a natural operation that preserves $2$-polarity and also preserves the condition of being a cograph. We do not know such an operation for $k > 2$, nevertheless we believe that the results and methods discussed here will also be useful for higher $k$.

math.CO↗

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

Proper Orientations of Planar Bipartite Graphs

An orientation of a graph $G$ is proper if any two adjacent vertices have different indegrees. The proper orientation number $\overrightarrowχ(G)$ of a graph $G$ is the minimum of the maximum indegree, taken over all proper orientations of $G$. In this paper, we show that a connected bipartite graph may be properly oriented even if we are only allowed to control the orientation of a specific set of edges, namely, the edges of a spanning tree and all the edges incident to one of its leaves. As a consequence of this result, we prove that 3-connected planar bipartite graphs have proper orientation number at most 6. Additionally, we give a short proof that $\overrightarrowχ(G) \leq 4$, when $G$ is a tree and this proof leads to a polynomial-time algorithm to proper orient trees within this bound.

math.CO↗