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In the ordinary version of the pursuit-evasion game "cops and robbers", a team of cops and a robber occupy vertices of a graph and alternately move along the graph's edges, with perfect information about each other. If a cop lands on the robber, the cops win; if the robber can evade the cops indefinitely, he wins. In the variant "lazy cops and robbers", the cops may only choose one member of their squad to make a move when it's their turn. The minimum number of cops (respectively lazy cops) required to catch the robber is called the "cop number" (resp. "lazy cop number") of G and is denoted $c(G)$ (resp. $c_L(G)$). Previous work by Beveridge at al. has shown that the Petersen graph is the unique graph on ten vertices with $c(G)=3$, and all graphs on nine or fewer vertices have $c(G)\leq 2$. (This was a self-contained mathematical proof of a result found by computational search by Baird and Bonato.) In this article, we prove a similar result for lazy cops, namely that the 3x3 rooks graph ($K_3\square K_3$) is the unique graph on nine vertices which requires three lazy cops, and a graph on eight or fewer vertices requires at most two lazy cops., Comment: 24 pages, 14 figures

Photosynthetic water oxidation is a fundamental process that sustains the biosphere. A Mn$_{4}$Ca cluster embedded in the photosystem II protein environment is responsible for the production of atmospheric oxygen. Here, time-resolved x-ray emission spectroscopy (XES) was used to observe the process of oxygen formation in real time. These experiments reveal that the oxygen evolution step, initiated by three sequential laser flashes, is accompanied by rapid (within 50 $\mu$s) changes to the Mn K$\beta$ XES spectrum. However, no oxidation of the Mn$_{4}$Ca core above the all Mn$^{\text{IV}}$ state was detected to precede O-O bond formation. A new mechanism featuring Mn$^{\text{IV}}$=O formation in the S$_{3}$ state is proposed to explain the spectroscopic results. This chemical formulation is consistent with the unique reactivity of the S$_{3}$ state and explains facilitation of the following S$_{3}$ to S$_{0}$ transition, resolving in part the kinetic limitations associated with O-O bond formation. In the proposed mechanism, O-O bond formation precedes transfer of the final (4$^{\text{th}}$) electron from the Mn$_{4}$Ca cluster, in agreement with experiment., Comment: 49 pages, includes main text (pgs. 1-16) and supplementary material (pgs. 17-49), 13 figures (4 main text, and 9 supplementary), 12 tables (2 main text, 10 supplementary)