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This paper revisits the 2-approximation algorithm for k-MST presented by Garg [9] in light of a recent paper of Paul et al. [14]. In the k-MST problem, the goal is to return a tree spanning k vertices of minimum total edge cost. Paul et al. [14] extend Garg's primal-dual subroutine to improve the approximation ratios for the budgeted prize-collecting traveling salesman and minimum spanning tree problems. We follow their algorithm and analysis to provide a cleaner version of Garg's result. Additionally, we introduce the novel concept of a kernel which allows an easier visualization of the stages of the algorithm and a clearer understanding of the pruning phase. Other notable updates include presenting a linear programming formulation of the k-MST problem, including pseudocode, replacing the coloring scheme used by Garg with the simpler concept of neutral sets, and providing an explicit potential function. 

Expensive instruments and complicated data processing are often required to discriminate solvents with similar structures and properties. Colorimetric sensors with high selectivity, low cost, and good portability are highly desirable to simplify such detection tasks. Herein, we report the fabrication of a photonic crystal sensor based on the self-assembled resorcinol formaldehyde (RF) hollow spheres to realize colorimetric sensing of polar solvents, including homologs and isomers based on the saturated diffusion time. The diffusion of solvent molecules through the photonic crystal film exhibits a unique three-step diffusion profile accompanied by a dynamic color change, as determined by the physicochemical properties of the solvent molecules and their interactions with the polymer shells, making it possible to accurately identify the solvent type based on the dynamic reflection spectra or visual perception. With its superior selectivity and sensitivity, this single-component colorimetric sensor represents a straightforward tool for convenient solvent detection and identification. 

Abstract Conventional templating synthesis confines the growth of seeds in rigid spaces to achieve faithful morphological replication. Herein, we explore the use of spherical shape‐deformable polymeric nanoshells to regulate the anisotropic growth of Ag nanoplates. The flexible shells deform adaptively to accommodate the initial overgrowth of the seeds but restrict the growth in the directions where the shells are fully stretched, eventually producing nanoplates with an unconventional circular profile. The diameter of the final Ag nanoplates can be precisely predicted by stretching and flattering the nanoshells into a plate‐like capsule while retaining their original internal surface area. Furthermore, unlike conventional templates, the polymer shells eventually turn themselves into a conformal coating that binds to the surface of the full‐grown Ag nanoplates and significantly enhances their stability against oxidative etching.