Given an n-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves straight-line planarity and consists of O(n) steps, which we prove is optimal in the worst case. Each step is a unidirectional linear morph, which means that every vertex moves at constant speed along a straight line, and the lines are parallel although the vertex speeds may differ. Thus we provide an efficient version of Cairns’ 1944 proof of the existence of straight-line planarity-preserving morphs for triangulated graphs, which required an exponential number of steps

The original notion of dimension for posets is due to Dushnik and Miller and has been studied extensively in the literature. Quite recently, there has been considerable interest in two variations of dimension known as Boolean dimension and local dimension. For a poset P , the Boolean dimension of P and the local dimension of P are both bounded from above by the dimension of P and can be considerably less. Our primary goal will be to study analogies and contrasts among these three parameters. As one example, it is known that the dimension of a poset is bounded as a function of its height and the tree-width of its cover graph. The Boolean dimension of a poset is bounded in terms of the tree-width of its cover graph, independent of its height. We show that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of height. We also prove that the local dimension of a poset is bounded in terms of the path-width of its cover graph. In several of our results, Ramsey theoretic methods will be applied.

Prediction of RNA base pairings yields insight into molecular structure, and therefore function. The most common methods predict an optimal structure under the standard thermodynamic model. One component of this model is the equation which governs the cost of branching, where three or more helical "arms" radiate out from a multiloop (also known as a junction). The multiloop initiation equation has three parameters; changing those values can significantly alter the predicted structure. We give a complete analysis of the prediction accuracy, stability, and robustness for all possible parameter combinations for a diverse set of tRNA sequences, and also for 5S rRNA. We find that the accuracy can often be substantially improved on a per sequence basis. However, simultaneous improvement within families, and most especially between families, remains a challenge.

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