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José Cáceres

Publications and source records attributed to José Cáceres.

15 recordsLinked to original sources
arXiv2026

The neighbourhood convexity

In this paper, we investigate the neighbourhood convexity ($n$-convexity) on graphs, a new finite convexity space grounded in the common closed neighbourhood closure operator. Unlike standard path-based graph convexities, $n$-convexity shows a non-canonical behaviour, giving rise to compelling structural properties and being almost never hereditary. Focusing on the properties of graphs that form $n$-convex geometries, a parity distinction emerges: an $n$-convex geometry contains a star vertex if and only if the number of its vertices is odd. Every odd-order $n$-convex geometry can be uniquely constructed by attaching a star vertex to an even-order one. We introduce the concept of quasi-stars (vertices of degree $\vert{}V\vert{}-2$) and prove a reduction property that allows systematically reducing an $n$-convex geometry by removing a pair of vertices, one of which is a quasi-star. Finally, we explore the connections between $n$-convexity and $P(G)$, the neighbourhood preorder, demonstrating that $n$-convex sets are upsets of $P(G)$ and that, in star-free $n$-convex geometries, quasi-stars correspond precisely to the maximal elements of $P(G)$. We complete our study by classifying quasi-threshold and threshold $n$-convex geometries.

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arXiv2026

Independent Locating-Dominating Sets in Pseudotrees

An ILD-set in a connected graph is a subset $S$ of vertices such that it is both independent and locating-dominating. The independent locating-dominating number of a graph G is the minimum cardinality of an ILD-set set of $G$. A well-known fact is that any graph with girth at least 5 has an ILD-set, but that is not clear for graphs with girth 3 and 4. In this work, we prove that there are graphs with no ILD-sets for any order $n\geq 9$ and girth 4, also showing some sufficient conditions for a bipartite graph to contain an ILD-set. Moreover, we focus our attention on trees and on unicyclic graphs, showing that every tree and every unicyclic graph contains ILD-sets, whenever in the latter case, it is twin-free. Finally, a number of bounds, realization theorems and algorithms to find an ILD-set in those families of graphs are provided.

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arXiv2026

The Borsuk number of a graph

The Borsuk problem asks for the smallest number of subsets with strictly smaller diameters into which any bounded set in the $d$-dimensional space can be decomposed. It is a classical problem in combinatorial geometry that has been subject of much attention over the years, and research on variants of the problem continues nowadays in a plethora of directions. In this work, we propose a formulation of the problem in the context of graphs. Depending on how the graph is partitioned, we consider two different settings dealing either with the usual notion of diameter in abstract graphs, or with the diameter in the context of continuous graphs, where all points along the edges, instead of only the vertices, must be taken into account when computing distances. We present complexity results, exact computations and upper bounds on the parameters associated to the problem.

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arXiv2025

Not every graph can be reconstructed from its boundary distance matrix

A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$ if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from $u$ than $v$. The boundary $\partial(G)$ of $G$ is the set of all of its boundary vertices. The boundary distance matrix $\hat{D}_G$ of a graph $G=([n],E)$ is the square matrix of order $κ$, being $κ$ the order of $\partial(G)$, such that for every $i,j\in \partial(G)$, $[\hat{D}_G]_{ij}=d_G(i,j)$. In a recent paper [doi.org/10.7151/dmgt.2567], it was shown that if a graph $G$ is either a block graph or a unicyclic graph, then $G$ is uniquely determined by the boundary distance matrix $\hat{D}_{G}$ of $G$, and it was also conjectured that this statement holds for every connected graph $G$, whenever both the order $n$ and the boundary (and thus also the boundary distance matrix) of $G$ are prefixed. After proving that this conjecture is true for several graph families, such as being of diameter 2, having order at most $n=6$ or being Ptolemaic, we show that this statement does not hold when considering, for example, either the family of split graphs of diameter 3 and order at least $n=10$ or the family of distance-hereditary graphs of order at least $n=8$.

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arXiv2024

Reconstructing a graph from the distance matrix of its boundary

A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$ if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from $u$ than $v$. The boundary $\partial(G)$ of $G$ is the set of all of its boundary vertices. The boundary distance matrix $\hat{D}_G$ of a graph $G=([n],E)$ is the square matrix of order $κ$, being $κ$ the order of $\partial(G)$, such that for every $i,j\in \partial(G)$, $[\hat{D}_G]_{ij}=d_G(i,j)$. Given a square matrix $\hat{B}$ of order $κ$, we prove under which conditions $\hat{B}$ is the distance matrix $\hat{D}_T$ of the set of leaves of a tree $T$, which is precisely its boundary. We show that if $G$ is either a block graph or a unicyclic graph, then $G$ is uniquely determined by the boundary distance matrix $\hat{D}_{G}$ of $G$ and we also conjecture that this statement holds for every connected graph $G$, whenever both the order $n$ and the boundary (and thus also the boundary distance matrix) of $G$ are prefixed. Moreover, an algorithm for reconstructing a 1-block graph (resp., a unicyclic graph) from its boundary distance matrix is given, whose time complexity in the worst case is $O(κn)$ (resp., $O(n^2)$).

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arXiv2018

General bounds on limited broadcast domination

Dominating broadcasting is a domination-type structure that models a transmission antenna network. In this paper, we study a limited version of this structure, that was proposed as a common framework for both broadcast and classical domination. In this limited version, the broadcast function is upper bounded by an integer $k$ and the minimum cost of such function is the dominating $k$-broadcast number. Our main result is a unified upper bound on this parameter for any value of $k$ in general graphs, in terms of both $k$ and the order of the graph. We also study the computational complexity of the associated decision problem.

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arXiv2017

Dominating 2-broadcast in graphs: complexity, bounds and extremal graphs

Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating $2$-broadcasts along with the associated parameter, the dominating $2$-broadcast number. We prove that computing the dominating $2$-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desired.

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arXiv2016

Shortcut sets for the locus of plane Euclidean networks

We study the problem of augmenting the locus $\mathcal{N}_{\ell}$ of a plane Euclidean network $\mathcal{N}$ by inserting iteratively a finite set of segments, called \emph{shortcut set}, while reducing the diameter of the locus of the resulting network. There are two main differences with the classical augmentation problems: the endpoints of the segments are allowed to be points of $\mathcal{N}_{\ell}$ as well as points of the previously inserted segments (instead of only vertices of $\mathcal{N}$), and the notion of diameter is adapted to the fact that we deal with $\mathcal{N}_{\ell}$ instead of $\mathcal{N}$. This increases enormously the hardness of the problem but also its possible practical applications to network design. Among other results, we characterize the existence of shortcut sets, compute them in polynomial time, and analyze the role of the convex hull of $\mathcal{N}_{\ell}$ when inserting a shortcut set. Our main results prove that, while the problem of minimizing the size of a shortcut set is NP-hard, one can always determine in polynomial time whether inserting only one segment suffices to reduce the diameter.

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arXiv2016

Quasi-efficient domination in grids

Domination of grids has been proved to be a demanding task and with the addition of independence it becomes more challenging. It is known that no grid with $m,n \geq 5$ has an efficient dominating set, also called perfect code, that is, an independent vertex set such that each vertex not in it has exactly one neighbor in that set. So it is interesting to study the existence of independent dominating sets for grids that allow at most two neighbors, such sets are called independent $[1,2]$-sets. In this paper we prove that every grid has an independent $[1,2]$-set, and we develop a dynamic programming algorithm using min-plus algebra that computes $i_{[1,2]}(P_m\Box P_n)$, the minimum cardinality of an independent $[1,2]$-set for the grid graph $P_m\square P_n$. We calculate $i_{[1,2]}(P_m\Box P_n)$ for $2\leq m\leq 13, n\geq m$ using this algorithm, meanwhile the parameter for grids with $14\leq m\leq n$ is obtained through a quasi-regular pattern that, in addition, provides an independent $[1,2]$-set of minimum size.

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arXiv2015

Perfect and quasiperfect domination in trees

A $k-$quasiperfect dominating set ($k\ge 1$) of a graph $G$ is a vertex subset $S$ such that every vertex not in $S$ is adjacent to at least one and at most k vertices in $S$. The cardinality of a minimum k-quasiperfect dominating set in $G$ is denoted by $γ_{\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept. The quasiperfect domination chain $γ_{\stackrel{}{11}}(G)\geγ_{\stackrel{}{12}}(G)\ge\dots\geγ_{\stackrel{}{1Δ}}(G)=γ(G)$, indicates what it is lost in size when you move towards a more perfect domination. We provide an upper bound for $γ_{\stackrel{}{1k}}(T)$ in any tree $T$ and trees achieving this bound are characterized. We prove that there exist trees satisfying all the possible equalities and inequalities in this chain and a linear algorithm for computing $γ_{\stackrel{}{1k}}(T)$ in any tree is presented.

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arXiv2014

On Perfect and Quasiperfect Domination in Graphs

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a $k$-quasiperfect dominating set (for $k\geq 1$) if every vertex not in $S$ is adjacent to at least one and at most $k$ vertices in $S$. The cardinality of a minimum $k$-quasiperfect dominating set in $G$ is denoted by $γ_ {\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers $ n \ge γ_ {\stackrel{}{11}}(G) \ge γ_ {\stackrel{}{12}}(G)\ge \ldots \ge γ_ {\stackrel{}{1Δ}}(G)=γ(G)$ in order to indicate how far is $G$ from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, $γ_ {\stackrel{}{12}}(G)=γ(G)$. Among them, one can find cographs, claw-free graphs and graphs with extremal values of $Δ(G)$.

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arXiv2012

Resolving sets for Johnson and Kneser graphs

A set of vertices $S$ in a graph $G$ is a {\em resolving set} for $G$ if, for any two vertices $u,v$, there exists $x\in S$ such that the distances $d(u,x) \neq d(v,x)$. In this paper, we consider the Johnson graphs $J(n,k)$ and Kneser graphs $K(n,k)$, and obtain various constructions of resolving sets for these graphs. As well as general constructions, we show that various interesting combinatorial objects can be used to obtain resolving sets in these graphs, including (for Johnson graphs) projective planes and symmetric designs, as well as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems and toroidal grids.

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arXiv2012

Locating dominating codes: Bounds and extremal cardinalities

In this work, two types of codes such that they both dominate and locate the vertices of a graph are studied. Those codes might be sets of detectors in a network or processors controlling a system whose set of responses should determine a malfunctioning processor or an intruder. Here, we present our more significant contributions on λ-codes and η-codes concerning concerning bounds, extremal values and realization theorems.

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arXiv2006

On the Metric Dimension of Cartesian Products of Graphs

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded.

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