Bounds for the Number of Independent and Dominating Sets in Trees

In this work, we investigate bounds on the number of independent sets in a graph and its complement, along with the corresponding question for number of dominating sets. Nordhaus and Gaddum gave bounds on χ(G)+χ(G) and χ(G) χ(G), where G is any graph on n vertices and χ(G) is the chromatic number of G. Nordhaus-Gaddum- type inequalities have been studied for many other graph invariants. In this work, we concentrate on i(G), the number of independent sets in G, and ∂(G), the number of dominating sets in G. We focus our attention on Nordhaus-Gaddum-type inequalities over trees on a fixed number of vertices. In particular, we give sharp upper and lower bounds on i(T )+ i(T ) where T is a tree on n vertices, improving bounds and proofs of Hu and Wei. We also give upper and lower bounds on i(G) + i(G) where G is a unicyclic graph on n vertices, again improving a result of Hu and Wei. Lastly, we investigate ∂(T )+ ∂(T ) where T is a tree on n vertices. We use a result of Wagner to give a lower bound and make a conjecture about an upper bound

Delayed colorectal cancer care during covid-19 pandemic (decor-19). Global perspective from an international survey

Background The widespread nature of coronavirus disease 2019 (COVID-19) has been unprecedented. We sought to analyze its global impact with a survey on colorectal cancer (CRC) care during the pandemic. Methods The impact of COVID-19 on preoperative assessment, elective surgery, and postoperative management of CRC patients was explored by a 35-item survey, which was distributed worldwide to members of surgical societies with an interest in CRC care. Respondents were divided into two comparator groups: 1) ‘delay’ group: CRC care affected by the pandemic; 2) ‘no delay’ group: unaltered CRC practice. Results A total of 1,051 respondents from 84 countries completed the survey. No substantial differences in demographics were found between the ‘delay’ (745, 70.9%) and ‘no delay’ (306, 29.1%) groups. Suspension of multidisciplinary team meetings, staff members quarantined or relocated to COVID-19 units, units fully dedicated to COVID-19 care, personal protective equipment not readily available were factors significantly associated to delays in endoscopy, radiology, surgery, histopathology and prolonged chemoradiation therapy-to-surgery intervals. In the ‘delay’ group, 48.9% of respondents reported a change in the initial surgical plan and 26.3% reported a shift from elective to urgent operations. Recovery of CRC care was associated with the status of the outbreak. Practicing in COVID-free units, no change in operative slots and staff members not relocated to COVID-19 units were statistically associated with unaltered CRC care in the ‘no delay’ group, while the geographical distribution was not. Conclusions Global changes in diagnostic and therapeutic CRC practices were evident. Changes were associated with differences in health-care delivery systems, hospital’s preparedness, resources availability, and local COVID-19 prevalence rather than geographical factors. Strategic planning is required to optimize CRC care