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CRISPR/Cas systems enable gene editing through the induction of site‐specific DNA double‐strand breaks (DSB). However, the nature of the induced modification highly depends on the mechanism used for DNA DSB repair. Non‐homologous end joining (NHEJ)‐mediated targeted mutagenesis induced by CRISPR/Cas is an already standardly applied tool, which can lead to various different kinds of mutations at a specific genomic site. Nevertheless, precise genome modification using homologous donor sequences is still challenging in plants. Applications depending on the less frequent homologous recombination (HR) require further improvements to create an attractive and efficient tool for general application in plants. Focusing on this issue, we developed the in planta gene targeting (ipGT) system, which is based on the simultaneous excision of a stably integrated, homologous donor sequence and the induction of a DSB within the target site. In recent years, several improvements were achieved enhancing gene targeting (GT) frequencies. After the successful application of Streptococcus pyogenes Cas9 (SpCas9) and Staphylococcus aureus Cas9 (SaCas9) for ipGT, we were able to further improve the system using Lachnospiraceae bacterium Cas12a (LbCas12a), which also enables cleavage in T‐rich regions. Most recently, we tested an improved, temperature‐tolerant version of LbCas12a (ttLbCas12a) for ipGT and were able to further increase GT efficiencies. Here, we describe the experimental procedure of the recently published ipGT system using ttLbCas12a in Arabidopsis thaliana in detail. © 2020 The Authors. Basic Protocol 1: Construction of CRISPR/ttLbCas12a expression vector to analyze ipGT efficiencies Basic Protocol 2: Achieving heritable GT plants

We introduce the novel concepts of local and union book embeddings, and, as the corresponding graph parameters, the local page number pn (G) and the union page number pn u (G). Both parameters are relaxations of the classical page number pn(G), and for every graph G we have pn (G) pn u (G) pn(G). While for pn(G) one minimizes the total number of pages in a book embedding of G, for pn (G) we instead minimize the number of pages incident to any one vertex, and for pn u (G) we instead minimize the size of a partition of G with each part being a vertex-disjoint union of crossing-free subgraphs. While pn (G) and pn u (G) are always within a multiplicative factor of 4, there is no bound on the classical page number pn(G) in terms of pn (G) or pn u (G). We show that local and union page numbers are closer related to the graph's density, while for the classical page number the graph's global structure can play a much more decisive role. We introduce tools to investigate local and union book embeddings in exemplary considerations of the class of all planar graphs and the class of graphs of tree-width k.As an incentive to pursue research in this new direction, we offer a list of intriguing open problems.

The page number of a directed acyclic graph G is the minimum k for which there is a topological ordering of G and a k-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We address the long-standing open problem asking for the largest page number among all upward planar graphs. We improve the best known lower bound to 5 and present the first asymptotic improvement over the trivial O(n) upper bound, where n denotes the number of vertices in G. Specifically, we first prove that the page number of every upward planar graph is bounded in terms of its width, as well as its height. We then combine both approaches to show that every n-vertex upward planar graph has page number O(n 2/3 log 2/3 (n)).

We prove that the stack-number of the strong product of three n-vertex paths is Θ(n 1/3 ). The best previously known upper bound was O(n). No non-trivial lower bound was known. This is the first explicit example of a graph family with bounded maximum degree and unbounded stack-number.The main tool used in our proof of the lower bound is the topological overlap theorem of Gromov. We actually prove a stronger result in terms of so-called triangulations of Cartesian products. We conclude that triangulations of three-dimensional Cartesian products of any sufficiently large connected graphs have large stack-number.The upper bound is a special case of a more general construction based on families of permutations derived from Hadamard matrices.The strong product of three paths is also the first example of a bounded degree graph with bounded queue-number and unbounded stack-number. A natural question that follows from our result is to determine the smallest ∆0 such that there exist a graph family with unbounded stack-number, bounded queue-number and maximum degree ∆0. We show that ∆0 ∈ {6, 7}.

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