No abstract

Given a directed graph D = (V, A) with a set of d specified vertices S = {s1, . . . , s d } ⊆ V and a function f : S → N where N denotes the set of natural numbers, we present a necessary and sufficient condition such that there existare rooted at si and each Ti,j spans the vertices from which si is reachable. This generalizes the result of Edmonds [2], i.e., the necessary and sufficient condition that for a directed graph D = (V, A) with a specified vertex s ∈ V , there are k arc-disjoint in-trees rooted at s each of which spans V . Furthermore, we extend another characterization of packing in-trees of Edmonds [1] to the one in our case.

In this paper, we consider the quickest flow problem in a network which consists of a directed graph with capacities and transit times on its arcs. We present an O(n log n) time algorithm for the quickest flow problem in a network of grid structure with uniform arc capacity which has a single sink where n is the number of vertices in the network.

This study aims to clarify the impact of the COVID-19 pandemic on home range. The home range is the area that individuals traverse in conducting their daily activities, such as working and shopping. In Japan, the central government declared the first state of emergency in April 2020. This study analyzed the panel data for mobile phone GPS location history from April 2019 to April 2020 in Ibaraki City, Osaka Metropolitan area. The study applied the minimum convex polygon method to analyze the data. The results show that the home range decreased significantly between April 2019 and April 2020. Specifically, the home range in 2020 decreased to approximately 50% of that in 2019 because of COVID-19 infection control measures, preventing people from traveling far from their homes and only allowing them to step outside for the bare minimum of daily activities and necessities. The results suggest that the emergency reduced people’s home ranges to the neighborhood scale. Therefore, it is necessary to consider designing new walkable neighborhood environments after the COVID-19 pandemic era.