The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families

We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as V(G), and the following terminology. Two vertices u,v is an element of V(G) are strongly resolved by a vertex w is an element of V(G), if there is a shortest w-v path containing u or a shortest w-u containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family F of graphs defined over a common vertex set V, a set S subset of V is an SSMG for F, if such set S is a strong metric generator for every graph G is an element of F. The simultaneous strong metric dimension of F is the minimum cardinality of any strong metric generator for F, and is denoted by Sds(F). The notion of simultaneous strong resolving graph of a graph family F is introduced in this work, and its usefulness in the study of Sds(F) is described. That is, it is proved that computing Sds(F) is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of F. Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature

The k-metric dimension of a graph

As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if the elements of any pair of different vertices of $G$ are distinguished by at least $k$ elements of $S$, i.e., for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. A connected graph $G$ is $k$-metric dimensional if $k$ is the largest integer such that there exists a $k$-metric basis for $G$. We give a necessary and sufficient condition for a graph to be $k$-metric dimensional and we obtain several results on the $k$-metric dimension

The Mighty Equine: The Influence of Titian and Rubens on the Equestrian Portraits of Velázquez

Diego Rodriguez de Silva y Velázquez (1599-1660) is considered by many to be one of the greatest artists Spain has ever produced. This essay explores the relationship between the painters Titian (ca. 1488-1576), Peter Paul Rubens (1577-1640), and Velázquez. In analyzing aspects of Velázquez\u27s equestrian portraiture of the family of Philip IV, circa 1635-1636, Isabel of Bourbon, Infante Baltasar Carlos, and especially that of the king himself, Philip IV, and comparing them with Titian\u27s Charles V at Mühlberg (ca. 1548) and The Cardinal-Infante Ferdinand at the Battle of Nördlingen (1634) by Rubens, it is possible to see stylistic similarities between the great masters. This research includes a copy of a lost Rubens work, Equestrian Portrait of Philip IV (1628), completed around 1645. Evidence is presented through a comparison of the painting with works by Rubens and Velázquez to suggest that the copy was primarily executed by a disciple of Velázquez, Juan Bautista Martínez de Mazo (ca. 1612-1667), but that the face of the king and the head of the horse were done by Velázquez himself. The event that proved pivotal to Velázquez can be traced to a visit by Rubens to the Spanish court of Philip IV in 1628-1629. A brief history of 17th-century Spain as it pertains to the Habsburg court is discussed. Although Titian is profoundly important to the development of Velázquez, Rubens appears to have been the bigger direct influence, both as an artist and as a member of the court, acting as the catalyst for Velázquez to take special notice of the works of Titian within the royal collections and encouraging the Spaniard to study in Italy. Titian became a major influence on him through his studies, which can be seen in the later equestrian portraits through the simplicity of compositions and the kingly virtues that the figures display through their royal bearing. Velázquez developed new skills on his travels, such as looser brushwork and a wider color palette, during his year and a half visit to Italy (1629-1631). Also, Velázquez broadened his subject matter to include mythological works. The painter was able to incorporate these new techniques into his existing style in order to paint in any manner that best suited his artistic purpose. Velázquez, Titian, and Rubens are three artists who are intertwined in their artistic careers. Titian inspired everyone who came into contact with his work; Rubens took notice and in turn presented an example for Velázquez to follow. This essay deals primarily with period surrounding the meeting between Rubens and Velázquez in 1628 and the creation of the equestrian portraits in the mid-1630s