arXiv subjects

María Inés Lopez Pujato

Publications and source records attributed to María Inés Lopez Pujato.

4 recordsLinked to original sources

arXiv2026

Released packing functions in graphs

We introduce and start the study of a variant of packing functions in graphs. Given a graph $G$ with vertex set $V$ and nonnegative integer vectors $\mathbf{k}=(k_v)_{v\in V}$, $\boldsymbol\ell=(l_v)_{v\in V}$ and $\mathbf{u}=(u_v)_{v\in V}$, a function $f : V \rightarrow \mathbb{Z}_0^+$ is a Released $( \mathbf{k}, \boldsymbol\ell, \mathbf{u})$-packing function of $G$ if $l_v\leq f(v)\leq u_v$ for every $v\in V$ and the sum of the values of $f$ over the closed neighborhood of vertices $v$ with $f(v) = u_v$ is at most $k_v$. The weight of $f$ is the value $f(V) = \sum_{v\in V} f(v)$. We study the associated decision problem (RPP), which asks, given $G$, $\mathbf{k}$, $\boldsymbol\ell$, $\mathbf{u}$ and an integer number $x$, whether $G$ admits a Released $( \mathbf{k}, \boldsymbol\ell, \mathbf{u})$-packing function of weight at least $x$. We relate RPP to the $r$-dependent set problem, derive several NP-hardness results, model RPP as a compact (polynomial in size) Integer Linear Program, and take the first steps of a polyhedral study.

math.CO↗

arXiv2023

On the proper interval completion problem within some chordal subclasses

Given a property (graph class) $Π$, a graph $G$, and an integer $k$, the \emph{$Π$-completion} problem consists in deciding whether we can turn $G$ into a graph with the property $Π$ by adding at most $k$ edges to $G$. The $Π$-completion problem is known to be NP-hard for general graphs when $Π$ is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem %when $Π$ is the class of proper interval graphs (PIG) within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class.

cs.DM↗

arXiv2022

Efficient algorithms for tuple domination on co-biconvex graphs and web graphs

A vertex in a graph dominates itself and each of its adjacent vertices. The \emph{$k$-tuple domination problem}, for a fixed positive integer $k$, is to find a minimum sized vertex subset in a given graph such that every vertex is dominated by at least $k$ vertices of this set. From the computational point of view, this problem is NP-hard. It follows from previous works by Bui-Xuan et al.~(2013) and by Belmonte et al.~(2013) -- in the context of locally checkable vertex subset problems in graph classes with quickly computable and bounded min-width -- that the $k$-tuple domination problem is solvable in time $\mathcal{O}(|V(G)|^{6k+4})$ in the class of circular-arc graphs. In this work, we develop faster algorithms for $k$-tuple domination in co-biconvex graphs and in web graphs, which are incomparable subclasses of concave-round graphs and thus of circular-arc graphs. On the one hand, we present an $\mathcal{O}(n^2)$-time algorithm for solving it for each $2\leq k\leq |U|+3$, where $U$ is the set of universal vertices and $n$ the total number of vertices of the input co-biconvex graph. On the other hand, the study of this problem on web graphs was already started by Argiroffo et al. (2010) from a polyhedral point of view only for the cases $k=2$ and $k=d(G)$, where $d(G)$ equals the degree of each vertex of the input web graph $G$. We complete this study for web graphs from an algorithmic point of view, by designing a linear-time algorithm based on the modular arithmetic for integer numbers. The algorithms presented in this work are mutually independent but both exploit the circular properties of the augmented adjacency matrices of each studied graph class.

math.CO↗

arXiv2018

Tuple domination on graphs with the consecutive-zeros property

The $k$-tuple domination problem, for a fixed positive integer $k$, is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least $k$ vertices in this set. The $k$-tuple domination is NP-hard even for chordal graphs. For the class of circular-arc graphs, its complexity remains open for $k\geq 2$. A $0,1$-matrix has the consecutive 0's property (C0P) for columns if there is a permutation of its rows that places the 0's consecutively in every column. Due to A. Tucker, graphs whose augmented adjancency matrix has the C0P for columns are circular-arc. In this work we study the $k$-tuple domination problem on graphs $G$ whose augmented adjacency matrix has the C0P for columns, for $ 2\leq k\leq |U|+3$, where $U$ is the set of universal vertices of $G$. From an algorithmic point of view, this takes linear time.

math.CO↗