Throttling for the game of Cops and Robbers on graphs

We consider the cop-throttling number of a graph $G$ for the game of Cops and Robbers, which is defined to be the minimum of $(k + \text{capt}_k(G))$, where $k$ is the number of cops and $\text{capt}_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games. We provide some tools for bounding the cop-throttling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are $O(\sqrt n)$.Comment: 22 pages, 4 figure

Optimizing the trade-off between number of cops and capture time in Cops and Robbers

The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games in which both players play optimally. In this paper, we construct a family of graphs having $th_c(G)= \Omega(n^{2/3})$, establish a sublinear upper bound on the cop throttling number, and show that the cop throttling number of chordal graphs is $O(\sqrt{n})$. We also introduce the product cop throttling number $th_c^{\times}(G)$ as a parameter that minimizes the person-hours used by the cops. This parameter extends the notion of speed-up that has been studied in the context of parallel processing and network decontamination. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph $G$, $th_c^{\times}=1+rad(G)$.Comment: 19 pages, 3 figure

Variability of the winter wind waves and swell in the North Atlantic and North Pacific as revealed by the Voluntary Observing Ship data

This paper analyses secular changes and interannual variability in the wind wave, swell, and significant wave height (SWH) characteristics over the North Atlantic and North Pacific on the basis of wind wave climatology derived from the visual wave observations of voluntary observing ship (VOS) officers. These data are available from the International Comprehensive Ocean–Atmosphere Data Set (ICOADS) collection of surface meteorological observations for 1958–2002, but require much more complicated preprocessing than standard meteorological variables such as sea level pressure, temperature, and wind. Visual VOS data allow for separate analysis of changes in wind sea and swell, as well as in significant wave height, which has been derived from wind sea and swell estimates. In both North Atlantic and North Pacific midlatitudes winter significant wave height shows a secular increase from 10 to 40 cm decade−1 during the last 45 yr. However, in the North Atlantic the patterns of trend changes for wind sea and swell are quite different from each other, showing opposite signs of changes in the northeast Atlantic. Trend patterns of wind sea, swell, and SWH in the North Pacific are more consistent with each other. Qualitatively the same conclusions hold for the analysis of interannual variability whose leading modes demonstrate noticeable differences for wind sea and swell. Statistical analysis shows that variability in wind sea is closely associated with the local wind speed, while swell changes can be driven by the variations in the cyclone counts, implying the importance of forcing frequency for the resulting changes in significant wave height. This mechanism of differences in variability patterns of wind sea and swell is likely more realistic than the northeastward propagation of swells from the regions from which the wind sea signal originates

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

Cop throttling number: Bounds, values, and variants

The cop throttling number thc(G) of a graph G for the game of Cops and Robbers is the minimum of k+captk(G), where k is the number of cops and captk(G) is the minimum number of rounds needed for k cops to capture the robber on G over all possible games in which both players play optimally. In this paper, we answer in the negative a question from [Breen et al., Throttling for the game of Cops and Robbers on graphs, {\em Discrete Math.}, 341 (2018) 2418--2430.] about whether the cop throttling number of any graph is O(n−−√) by constructing a family of graphs having thc(G)=Ω(n2/3). We establish a sublinear upper bound on the cop throttling number and show that the cop throttling number of chordal graphs is O(n−−√). We also introduce the product cop throttling number th×c(G) as a parameter that minimizes the person-hours used by the cops. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph G, th×c(G)=1+rad(G)

Indocyanine green fluorescence image processing techniques for breast cancer macroscopic demarcation

Re-operation due to disease being inadvertently close to the resection margin is a major challenge in breast conserving surgery (BCS). Indocyanine green (ICG) fluorescence imaging could be used to visualize the tumor boundaries and help surgeons resect disease more efficiently. In this work, ICG fluorescence and color images were acquired with a custom-built camera system from 40 patients treated with BCS. Images were acquired from the tumor in-situ, surgical cavity post-excision, freshly excised tumor and histopathology tumour grossing. Fluorescence image intensity and texture were used as individual or combined predictors in both logistic regression (LR) and support vector machine models to predict the tumor extent. ICG fluorescence spectra in formalin-fixed histopathology grossing tumor were acquired and analyzed. Our results showed that ICG remains in the tissue after formalin fixation. Therefore, tissue imaging could be validated in freshly excised and in formalin-fixed grossing tumor. The trained LR model with combined fluorescence intensity (pixel values) and texture (slope of power spectral density curve) identified the tumor’s extent in the grossing images with pixel-level resolution and sensitivity, specificity of 0.75 ± 0.3, 0.89 ± 0.2.This model was applied on tumor in-situ and surgical cavity (post-excision) images to predict tumor presence

Strong Structural Controllability and Zero Forcing

In this chapter, we study controllability and output controllability of systems defined over graphs. Specifically, we consider a family of state-space systems, where the state matrix of each system has a zero/non-zero structure that is determined by a given directed graph. Within this setup, we investigate under which conditions all systems in this family are controllable, a property referred to as strong structural controllability. Moreover, we are interested in conditions for strong structural output controllability. We will show that the graph-theoretic concept of zero forcing is instrumental in these problems. In particular, as our first contribution, we prove necessary and sufficient conditions for strong structural controllability in terms of so-called zero forcing sets. Second, we show that zero forcing sets can also be used to state both a necessary and a sufficient condition for strong structural output controllability. In addition to these main results, we include interesting results on the controllability of subfamilies of systems and on the problem of leader selection.</p

Nintedanib for non-IPF progressive pulmonary fibrosis: 12-month outcome data from a real-world multicentre observational study

Background: Nintedanib slows lung function decline for patients with non-IPF progressive pulmonary fibrosis (PPF) in clinical trials, but the real-world safety and efficacy are not known. Methods: In this retrospective cohort study, standardised data was collected across 8 UK centres from patients in whom nintedanib was initiated for PPF between 2019 and 2020 through an early access programme. Rate of lung function change in the 12 months pre- and post-nintedanib initiation was the primary analysis. Symptoms, drug safety, tolerability, and stratification by interstitial lung disease (ILD) subtype and CT pattern were secondary analyses. Results: 126 patients were included; 67(53%) females, mean age 60(±13) years. At initiation of nintedanib, mean FVC was 1.87 L (58%) and DLco 32.7% predicted. 68% of patients were prescribed prednisolone (median dose 10 mg) and 69% prescribed a steroid sparing agent. In the 12 months after nintedanib initiation, lung function decline was significantly lower than in the preceding 12 months; FVC −88.8 ml versus −239.9 ml respectively, (p=0.004) and absolute decline in DLco −2.1% versus −6.1% respectively; (p=0.004). Response to nintedanib was consistent in sensitivity and secondary analyses. 89/126 (71%) of patients reported side effects but 86 of the surviving 108 patients (80%) were still taking nintedanib at 12 months with patients reporting a reduced perception of symptom decline. There were no serious adverse events. Conclusion: In PPF, the real-world efficacy of nintedanib replicated that of clinical trials, significantly attenuating lung function decline. Despite the severity of disease, nintedanib was safe and well tolerated in this real-world multicentre stud

Germline sequencing in men with metastatic castration-resistant prostate cancer from the BARCODE2 study reveals a wide range of pathogenic variants in DNA repair genes

Abstract Background The presence of germline mutations plays an increasingly important role in risk assessment and treatment of prostate cancer (PrCa). Screening for high-risk mutations in subsets of patients is becoming routine. We explore the prevalence of germline genetic mutations in men with metastatic castration-resistant prostate cancer (mCRPC) recruited to the BARCODE2 trial. Methods The BARCODE2 trial is a two-part study investigating the response to carboplatin chemotherapy in mCRPC patients carrying a germline variant in a DNA repair gene (DRG). We report interim data from Part 1, in which participants are recruited for germline genetic testing using a customised next-generation sequencing panel consisting of 115 genes. Results These interim results (N = 220) demonstrate a similar frequency of germline DRG variants in mCRPC patients compared with previously published data (15% detection rate). No significant clinical differences were identified between all carriers and non-carriers, though BRCA2/ATM carriers were found to have a shorter time to mCRPC diagnosis. Conclusions Germline pathogenic/likely pathogenic (P/LP) variants in BRCA2 and ATM genes are associated with a shorter time to progression and rarer P/LP variants in other DRG genes may play a role in mCRPC. This justifies the use of routine screening of men with advanced PrCa for germline variants and supports the need for an expanded panel test. </jats:sec