A Comparison between the Zero Forcing Number and the Strong Metric Dimension of Graphs

The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)-S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. The \emph{strong metric dimension}, $sdim(G)$, of a graph $G$ is the minimum among cardinalities of all strong resolving sets: $W \subseteq V(G)$ is a \emph{strong resolving set} of $G$ if for any $u, v \in V(G)$, there exists an $x \in W$ such that either $u$ lies on an $x-v$ geodesic or $v$ lies on an $x-u$ geodesic. In this paper, we prove that $Z(G) \le sdim(G)+3r(G)$ for a connected graph $G$, where $r(G)$ is the cycle rank of $G$. Further, we prove the sharp bound $Z(G) \leq sdim(G)$ when $G$ is a tree or a unicyclic graph, and we characterize trees $T$ attaining $Z(T)=sdim(T)$. It is easy to see that $sdim(T+e)-sdim(T)$ can be arbitrarily large for a tree $T$; we prove that $sdim(T+e) \ge sdim(T)-2$ and show that the bound is sharp.Comment: 8 pages, 5 figure

On vector spaces with distinguished subspaces and redundant base

Abstract Let V , W be finite dimensional vector spaces over a field K, each with n distin- guished subspaces, with a dimension-preserving correspondence between intersec- tions. When does this guarantee the existence of an isomorphism between V and W matching corresponding subspaces? The setting where it happens requires that the distinguished subspaces be generated by subsets of a given redundant base of the space; this gives rise to a (0,1)-incidence table called tent, an object which occurs in the study of Butler B(1)-groups

Transcatheter closure of complex iatrogenic ventricular septal defect: A case report

Background: Iatrogenic membranous ventricular septal defects (VSDs) are rare complications of cardiothoracic surgery, such as septal myectomy for hypertrophic obstructive cardiomyopathy (HOCM). Transcatheter closure is considered an appealing alternative to surgery, given the increased mortality associated with repeated surgical procedures, but reports are extremely limited. Case summary: We herein report the case of a 63-year-old woman with HOCM who underwent successful percutaneous closure of an iatrogenic VSD after septal myectomy. Two percutaneous techniques are discussed, namely the 'muscular anchoring' and the 'buddy wire delivery', aimed at increasing support and providing stability to the system during percutaneous intervention. Discussion: Transcatheter closure represents an attractive minimally invasive approach for the management of symptomatic iatrogenic VSDs. The new techniques described could help operators to cross tortuous and tunnelled defects and to deploy closure devices in case of complex VSD anatomy

On the Power Domination Number of de Bruijn and Kautz Digraphs

Let G=(V,A) be a directed graph, and let S⊆V be a set of vertices. Let the sequence S=S₀⊆S₁⊆S₂⊆⋯ be defined as follows: S₁ is obtained from S₀ by adding all out-neighbors of vertices in S₀. For k⩾2, Sₖ is obtained from Sₖ₋₁ by adding all vertices w such that for some vertex v∈Sₖ₋₁, w is the unique out-neighbor of v in V∖Sₖ₋₁. We set M(S)=S₀∪S₁∪⋯, and call S a power dominating set for G if M(S)=V(G). The minimum cardinality of such a set is called the power domination number of G. In this paper, we determine the power domination numbers of de Bruijn and Kautz digraphs

A five-year survey of tospoviruses infecting vegetable crops in main producing regions of Brazil.

The present work describes a five-year survey (2010?2014) of tospoviruses infecting these vegetable crops under Brazilian conditions. A total of 318 samples from tomato (Solanum lycopersicon L. ? 285), lettuce (Lactuca sativa L. ? 14), pepper (Capsicum L. species ? 11), gilo (Solanum aethiopicum var. gilo Raddi ? 5) and pigweed (Amaranthus L. species ? 5) were collected from plants displaying typical tospovirus symptoms

On the minimum rank of not necessarily symmetric matrices : a preliminary study

The minimum rank of a directed graph G is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in G and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i _= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem