Size reconstructibility of graphs

The deck of a graph $G$ is given by the multiset of (unlabelled) subgraphs $\{G-v:v\in V(G)\}$. The subgraphs $G-v$ are referred to as the cards of $G$. Brown and Fenner recently showed that, for $n\geq29$, the number of edges of a graph $G$ can be computed from any deck missing 2 cards. We show that, for sufficiently large $n$, the number of edges can be computed from any deck missing at most $\frac1{20}\sqrt{n}$ cards.Comment: 15 page

Labeling Schemes for Bounded Degree Graphs

We investigate adjacency labeling schemes for graphs of bounded degree $\Delta = O(1)$. In particular, we present an optimal (up to an additive constant) $\log n + O(1)$ adjacency labeling scheme for bounded degree trees. The latter scheme is derived from a labeling scheme for bounded degree outerplanar graphs. Our results complement a similar bound recently obtained for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new insights for closing the long standing gap for adjacency in trees [Alstrup and Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree planar graphs. Finally, we use combinatorial number systems and present an improved adjacency labeling schemes for graphs of bounded degree $\Delta$ with $(e+1)\sqrt{n} < \Delta \leq n/5$

Near-Optimal Computation of Runs over General Alphabet via Non-Crossing LCE Queries

Longest common extension queries (LCE queries) and runs are ubiquitous in algorithmic stringology. Linear-time algorithms computing runs and preprocessing for constant-time LCE queries have been known for over a decade. However, these algorithms assume a linearly-sortable integer alphabet. A recent breakthrough paper by Bannai et.\ al.\ (SODA 2015) showed a link between the two notions: all the runs in a string can be computed via a linear number of LCE queries. The first to consider these problems over a general ordered alphabet was Kosolobov (\emph{Inf.\ Process.\ Lett.}, 2016), who presented an $O(n (\log n)^{2/3})$-time algorithm for answering $O(n)$ LCE queries. This result was improved by Gawrychowski et.\ al.\ (accepted to CPM 2016) to $O(n \log \log n)$ time. In this work we note a special \emph{non-crossing} property of LCE queries asked in the runs computation. We show that any $n$ such non-crossing queries can be answered on-line in $O(n \alpha(n))$ time, which yields an $O(n \alpha(n))$-time algorithm for computing runs

Aram Dorsum: an extensive mid-Noachian age fluvial depositional system in Arabia Terra, Mars

A major debate in Mars science is the nature of the early Mars climate, and the availability of precipitation and runoff. Observations of relict erosional valley networks have been proposed as evidence for extensive surface run‐off around the Noachian‐Hesperian boundary. However, these valley networks only provide a time‐integrated record of landscape evolution and thus the timing, relative timescales and intensity of aqueous activity required to erode the valleys remain unknown. Here, we investigate an ancient fluvial sedimentary system in western Arabia Terra, now preserved in positive relief. This ridge, ‘Aram Dorsum’, is flat‐topped, branching, ~ 85 km long, and particularly well‐preserved. We show that Aram Dorsum was an aggradational alluvial system and that the existing ridge was once a large river channel‐belt set in extensive flood plains, many of which are still preserved. Smaller, palaeochannel‐belts feed the main system; their setting and network pattern suggest a distributed source of water. The alluvial succession is up to 60 m thick, suggesting a formation time of 105 to 107 years by analogy to Earth. Our observations are consistent with Aram Dorsum having formed by long‐lived flows of water, sourced both locally, and regionally as part of a wider alluvial system in Arabia Terra. This suggests frequent or seasonal precipitation as the source of water. Correlating our observations with previous regional‐scale mapping shows that Aram Dorsum formed in the mid‐Noachian, making it one of the oldest fluvial systems described on Mars and indicating climatic conditions that sustained surface river flows on early Mars

Computed tomography attenuation of periaortic adipose tissue in abdominal aortic aneurysms

Purpose: To assess periaortic adipose tissue attenuation on CT angiography indifferent abdominal aortic aneurysm disease states.Materials and Methods: In a retrospective observational study from January 2018 to December 2022, periaortic adipose tissue attenuation was assessed on CT angiography in patients with asymptomatic or symptomatic (including rupture) abdominal aortic aneurysms, and control individuals without aneurysms. Adipose tissue attenuation was measured using semi-automated software in periaortic aneurysmal and non-aneurysmal segments of the abdominal aorta, and in subcutaneous and visceral adipose tissue. Periaortic adipose tissue attenuation values between the three groups was assessed using Students t-test and Wilcoxon rank sum test followed by a multi-regression model.Results: Eighty-eight individuals (median age, 70 [IQR, 65-78] years; 78 male and 10 female) were included: 70 patients with abdominal aortic aneurysms (40 asymptomatic and 30 symptomatic including 24 with rupture), and 18 controls. There was no evidence of differences in the periaortic adipose tissue attenuation in the aneurysmal segment in asymptomatic patients versus controls ((-81.44±7 versus -83.27±9 HU, Hounsfield units, P=0.43) and attenuation in non-aneurysmal segments between asymptomatic patients versus controls (-75.43±8 versus -78.81±6 HU, P=0.08). However, symptomatic patients demonstrated higher periaortic adipose tissue attenuation in both aneurysmal (-57.85±7 HU, P&lt;0.0001) and non-aneurysmal segments (-58.16±8 HU, P&lt;0.0001) when compared with the other two groups.Conclusions: Periaortic adipose tissue CT attenuation was not increased in stableabdominal aortic aneurysm disease. There was a generalised increase in attenuation in patients with symptomatic disease, likely reflecting the systemic consequences of acute rupture

The generalized 3-edge-connectivity of lexicographic product graphs

The generalized $k$-edge-connectivity $\lambda_k(G)$ of a graph $G$ is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs $G$ and $H$, denoted by $G\circ H$, is an important graph product. In this paper, we mainly study the generalized 3-edge-connectivity of $G \circ H$, and get upper and lower bounds of $\lambda_3(G \circ H)$. Moreover, all bounds are sharp.Comment: 14 page

A Two-Phase, Single Cohort Study of COVID-19 Antibody Sera-Surveillance

Objective: The infectious corona virus, COVID-19 has high case mortality in those whom suffer with severe symptoms requiring hospitalisation. A major problem associated with COVID-19 is the spread of infection by a-symptomatic carriers, or those with mild symptoms. We aim to determine the antibody prevalence in a professional Welsh cohort and begin to explore the longevity of COVID-19 antibodies. Methods: 739 Cardiff Metropolitan University staff members took part in an observational study to determine the prevalence of COVID-19 antibodies in a two-phase, single cohort study. All participants were tested for IgM and IgG antibodies against COVID-19 using a lateral flow detection assay. Venous blood samples from positive participants and a randomly selected negative population were collected to confirm antibody titre, using two gold standard immunoassays, carried out independently by the Specialist Virology Centre, UHW. Results: 3.65% of the population tested positive for antibodies against COVID-19, with a higher prevalence seen in male participants (5% vs. 2.73% of females). In addition to gender, both pre-existing asthma and age were key determinants in antibody positivity. 78.26% retained antibodies at the 3 months follow up test. 36.36% of females lost antibody positivity between the 3 - and 6 - month time points compared with 8.3% of males. Lateral flow antibody testing was shown to have 96% sensitivity and 95% specificity compared with standard tests. Conclusion: We conclude that prevalence of COVID-19 antibodies is evident in the asymptomatic population, and in 78.26% of those initially antibody positive prevails at approximately 6 months from perceived time of exposure. Males are 4 times more likely to retain antibodies for longer than females