arXiv subjects

Julio Araujo

Publications and source records attributed to Julio Araujo.

4 recordsLinked to original sources

arXiv2023

The iteration time and the general position number in graph convexities

In this paper, we study two graph convexity parameters: iteration time and general position number. The iteration time was defined in 1981 in the geodesic convexity, but its computational complexity was so far open. The general position number was defined in the geodesic convexity and proved NP-hard in 2018. We extend these parameters to any graph convexity and prove that the iteration number is NP-hard in the P3 convexity. We use this result to prove that the iteration time is also NP-hard in the geodesic convexity even in graphs with diameter two, a long standing open question. These results are also important since they are the last two missing NP-hardness results regarding the ten most studied graph convexity parameters in the geodesic and P3 convexities. We also prove that the general position number of the monophonic convexity is W[1]-hard (parameterized by the size of the solution) and $n^{1-\varepsilon}$-inapproximable in polynomial time for any $\varepsilon>0$ unless P=NP, even in graphs with diameter two. Finally, we also obtain FPT results on the general position number in the P3 convexity and we prove that it is W[1]-hard (parameterized by the size of the solution).

cs.CC↗

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.

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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↗

arXiv2015

On the Existence of Tree Backbones that Realize the Chromatic Number on a Backbone Coloring

A proper $k$-coloring of a graph $G=(V,E)$ is a function $c: V(G)\to \{1,\ldots,k\}$ such that $c(u)\neq c(v)$, for every $uv\in E(G)$. The chromatic number $χ(G)$ is the minimum $k$ such that there exists a proper $k$-coloring of $G$. Given a spanning subgraph $H$ of $G$, a $q$-backbone $k$-coloring of $(G,H)$ is a proper $k$-coloring $c$ of $V(G)$ such that $\lvert c(u)-c(v)\rvert \ge q$, for every edge $uv\in E(H)$. The $q$-backbone chromatic number $BBC_q(G,H)$ is the smallest $k$ for which there exists a $q$-backbone $k$-coloring of $(G,H)$. In this work, we show that every connected graph $G$ has a generating tree $T$ such that $BBC_q(G,T) = \max\{χ(G),\left\lceil\frac{χ(G)}{2}\right\rceil+q\}$, and that this value is the best possible. As a direct consequence, we get that every connected graph $G$ has a spanning tree $T$ for which $BBC_2(G,T)=χ(G)$, if $χ(G)\ge 4$, or $BBC_2(G,T)=χ(G)+1$, otherwise. Thus, by applying the Four Color Theorem, we have that every connected nonbipartite planar graph $G$ has a spanning tree $T$ such that $BBC_2(G,T)=4$. This settles a question by Wang, Bu, Montassier and Raspaud (2012), and generalizes a number of previous partial results to their question.

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