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

Sustainment of the TeleSleep program for rural veterans

BackgroundIn fiscal year 2021, the Veterans Health Administration (VHA) provided care for sleep disorders to 599,966 Veterans, including 189,932 rural Veterans. To further improve rural access, the VA Office of Rural Health developed the TeleSleep Enterprise-Wide Initiative (EWI). TeleSleep's telemedicine strategies include tests for sleep apnea at the Veteran's home rather than in a sleep lab; Clinical Video Telehealth applications; and other forms of virtual care. In 2017 and 2020, VHA provided 3-year start-up funding to launch new TeleSleep programs at rural-serving VA medical facilities.MethodsIn early 2022, we surveyed leaders of 24 sites that received TeleSleep funding to identify successes, failures, facilitators, and barriers relevant to sustaining TeleSleep implementations upon expiration of startup funding. We tabulated frequencies on the multiple choice questions in the survey, and, using the survey's critical incident framework, summarized the responses to open-ended questions. TeleSleep program leaders discussed the responses and synthesized recommendations for improvement.Results18 sites reported sustainment, while six were “on track.” Sustainment involved medical centers or regional entities incorporating TeleSleep into their budgets. Facilitators included: demonstrating value; aligning with local priorities; and collaborating with spoke sites serving rural Veterans. Barriers included: misalignment with local priorities; and hiring delays. COVID was a facilitator, as it stimulated adoption of telehealth practices; and also a barrier, as it consumed attention and resources. Recommendations included: longer startup funding; dedicated funding for human resources to accelerate hiring; funders communicating with local facility leaders regarding how TeleSleep aligns with organizational priorities; hiring into job classifications aligned with market pay; and obtaining, from finance departments, projections and outcomes for the return on investment in TeleSleep

Combinatorial distance geometry in normed spaces

We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as minimum spanning trees and Steiner minimum trees. In particular, we discuss translative kissing (or Hadwiger) numbers, equilateral sets, and the Borsuk problem in normed spaces. We show how to use the angular measure of Peter Brass to prove various statements about Hadwiger and blocking numbers of convex bodies in the plane, including some new results. We also include some new results on thin cones and their application to distinct distances and other combinatorial problems for normed spaces