arXiv subjects

Víctor Franco-Sánchez

Publications and source records attributed to Víctor Franco-Sánchez.

2 recordsLinked to original sources

arXiv2026

Metric representations by minimal graphs

A resolving set in a graph $G$ is a vertex subset $W= \{ω^1, \dots, ω^n\} \subseteq V(G)$ such that each $u \in V(G)$ can be uniquely identified by the vector $r(u \vert W) = (d(u,ω^1), \dots, d(u,ω^n))$ of metric coordinates of $u$ with respect to $W$. The reverse problem of identifying the vector sets that are a set of coordinates of some graph provides the concept of realizable vector set $S \subset \mathbb{Z}^n$ by a pair $(G, W)$ meaning that $S=\{ r(u\vert W)\colon u\in V(G)\}$ with $W$ a resolving set of the graph $G$. Here we focus on the minimality of the realizations of vector sets with respect to their edge sets. On the one hand, we study conditions under which it is possible to remove an edge from the graph and keep the realizability condition. This provides a method for finding minimal realizations, as well as allowing us to characterize uniquely realizable vector sets. On the other hand, we prove that the decision problem of realizing a vector set by a graph with a given number of edges is an NP-complete problem. Finally, we characterize the vector sets that are realizable by a tree and, furthermore, we study the case in which such a realization is the only one.

math.CO↗

arXiv2026

Swap distance minimization beyond entropy minimization in word order variation

Consider a linguistic structure formed by $n$ elements, for instance, subject, direct object and verb ($n=3$) or subject, direct object, indirect object and verb ($n=4$). We investigate whether the frequency of the $n!$ possible orders is constrained by two principles. First, entropy minimization, a principle that has been suggested to shape natural communication systems at distinct levels of organization. Second, swap distance minimization, namely a preference for word orders that require fewer swaps of adjacent elements to be produced from a source order. We present average swap distance, a novel score for research on swap distance minimization. We find strong evidence of pressure for entropy minimization and swap distance minimization with respect to a die rolling experiment in distinct linguistic structures with $n=3$ or $n=4$. Evidence with respect to a Polya urn process is strong for $n=4$ but weaker for $n=3$. We still find evidence consistent with the action of swap distance minimization when word order frequencies are shuffled, indicating that swap distance minimization effects are beyond pressure to reduce word order entropy.

cs.CL↗