Two-Player Graph Pebbling

Given a graph G with pebbles on the vertices, we define a pebbling move as removing two pebbles from a vertex u, placing one pebble on a neighbor v, and discarding the other pebble, like a toll. The pebbling number n(G) is the least number of pebbles needed so that every arrangement of n(G) pebbles can place a pebble on any vertex through a sequence of pebbling moves. We introduce a new variation on graph pebbling called two-player pebbling. In this, players called the mover and the defender alternate moves, with the stipulation that the defender cannot reverse the previous move. The mover wins only if they can place a pebble on a specified vertex and the defender wins if the mover cannot. We define n(G), analogously, as the minimum number of pebbles such that given every configuration of the n(G) pebbles and every specified vertex r, the mover has a winning strategy. First, we will investigate upper bounds for n(G) on various classes of graphs and find a certain structure for which the defender has a winning strategy, no matter how many pebbles are in a configuration. Then, we characterize winning configurations for both players on a special class of diameter 2 graphs. Finally, we show winning configurations for the mover on paths using a recursive argument

A Pebbling Game on Powers of Paths

Two Player Graph Pebbling is an extension of graph pebbling. Players Mover and Defender use pebbling moves, the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex, to win. If a specified vertex has a pebble on it, then Mover wins. If a specified vertex is pebble-free and there are no more valid pebbling moves, then Defender wins. The Two-Player Pebbling Number of a graph G, η(G), is the minimum m such that for every arrangement of m pebbles and for any specified vertex, Mover can win. We specify the winning player for powers of a path

On Two-Player Pebbling

Graph pebbling can be extended to a two-player game on a graph G, called Two-Player Graph Pebbling, with players Mover and Defender. The players each use pebbling moves, the act of removing two pebbles from one vertex and placing one of the pebbles on an adjacent vertex, to win. Mover wins if they can place a pebble on a specified vertex. Defender wins if the specified vertex is pebble-free and there are no more pebbling moves on the vertices of G. The Two-Player Pebbling Number of a graph G, η(G), is the minimum m such that for every arrangement of m pebbles and for any specified vertex, Mover can win. We specify the winning player for paths, cycles, and the join of certain graphs

Clinical phenotypes and outcomes in children with multisystem inflammatory syndrome across SARS-CoV-2 variant eras: a multinational study from the 4CE consortiumResearch in context

Summary: Background: Multisystem inflammatory syndrome in children (MIS-C) is a severe complication of SARS-CoV-2 infection. It remains unclear how MIS-C phenotypes vary across SARS-CoV-2 variants. We aimed to investigate clinical characteristics and outcomes of MIS-C across SARS-CoV-2 eras. Methods: We performed a multicentre observational retrospective study including seven paediatric hospitals in four countries (France, Spain, U.K., and U.S.). All consecutive confirmed patients with MIS-C hospitalised between February 1st, 2020, and May 31st, 2022, were included. Electronic Health Records (EHR) data were used to calculate pooled risk differences (RD) and effect sizes (ES) at site level, using Alpha as reference. Meta-analysis was used to pool data across sites. Findings: Of 598 patients with MIS-C (61% male, 39% female; mean age 9.7 years [SD 4.5]), 383 (64%) were admitted in the Alpha era, 111 (19%) in the Delta era, and 104 (17%) in the Omicron era. Compared with patients admitted in the Alpha era, those admitted in the Delta era were younger (ES −1.18 years [95% CI −2.05, −0.32]), had fewer respiratory symptoms (RD −0.15 [95% CI −0.33, −0.04]), less frequent non-cardiogenic shock or systemic inflammatory response syndrome (SIRS) (RD −0.35 [95% CI −0.64, −0.07]), lower lymphocyte count (ES −0.16 × 109/uL [95% CI −0.30, −0.01]), lower C-reactive protein (ES −28.5 mg/L [95% CI −46.3, −10.7]), and lower troponin (ES −0.14 ng/mL [95% CI −0.26, −0.03]). Patients admitted in the Omicron versus Alpha eras were younger (ES −1.6 years [95% CI −2.5, −0.8]), had less frequent SIRS (RD −0.18 [95% CI −0.30, −0.05]), lower lymphocyte count (ES −0.39 × 109/uL [95% CI −0.52, −0.25]), lower troponin (ES −0.16 ng/mL [95% CI −0.30, −0.01]) and less frequently received anticoagulation therapy (RD −0.19 [95% CI −0.37, −0.04]). Length of hospitalization was shorter in the Delta versus Alpha eras (−1.3 days [95% CI −2.3, −0.4]). Interpretation: Our study suggested that MIS-C clinical phenotypes varied across SARS-CoV-2 eras, with patients in Delta and Omicron eras being younger and less sick. EHR data can be effectively leveraged to identify rare complications of pandemic diseases and their variation over time. Funding: None