arXiv subjects

Fernando Esteban Contreras-Mendoza

Publications and source records attributed to Fernando Esteban Contreras-Mendoza.

3 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↗

arXiv2024

$2$-polarity and algorithmic aspects of polarity variants on cograph superclasses

A graph $G$ is said to be an $(s, k)$-polar graph if its vertex set admits a partition $(A, B)$ such that $A$ and $B$ induce, respectively, a complete $s$-partite graph and the disjoint union of at most $k$ complete graphs. Polar graphs and monopolar graphs are defined as $(\infty, \infty)$- and $(1, \infty)$-polar graphs, respectively, and unipolar graphs are those graphs with a polar partition $(A, B)$ such that $A$ is a clique. The problems of deciding whether an arbitrary graph is a polar graph or a monopolar graph are known to be NP-complete. In contrast, deciding whether a graph is a unipolar graph can be done in polynomial time. In this work we prove that the three previous problems can be solved in linear time on the classes of $P_4$-sparse and $P_4$-extendible graphs, generalizing analogous results previously known for cographs. Additionally, we provide finite forbidden subgraph characterizations for $(2,2)$-polar graphs on $P_4$-sparse and $P_4$-extendible graphs, also generalizing analogous results recently obtained for the class of cographs.

math.CO↗

arXiv2022

Minimal obstructions for polarity, monopolarity, unipolarity and $(s,1)$-polarity in generalizations of cographs

It is known that every hereditary property can be characterized by finitely many minimal obstructions when restricted to either the class of cographs or the class of $P_4$-reducible graphs. In this work, we prove that also when restricted to the classes of $P_4$-sparse graphs and $P_4$-extendible graphs (both of which extend $P_4$-reducible graphs) every hereditary property can be characterized by finitely many minimal obstructions. We present complete lists of $P_4$-sparse and $P_4$-extendible minimal obstructions for polarity, monopolarity, unipolarity, and $(s,1)$-polarity, where $s$ is a positive integer. In parallel to the case of $P_4$-reducible graphs, all the $P_4$-sparse minimal obstructions for these hereditary properties are cographs.

math.CO↗