We combine two fundamental optimization problems related to the construction of phylogenetic trees called maximum rooted triplets consistency and minimally resolved supertree into a new problem, which we call q-maximum rooted triplets consistency ( q -MAXRTC). It takes as input a set R of rooted, binary phylogenetic trees with three leaves each and asks for a phylogenetic tree with exactly q internal nodes that contains the largest possible number of trees from R . We prove that q -MAXRTC is NP-hard to approximate within a constant, develop polynomial-time approximation algorithms for different values of q , and show experimentally that representing a phylogenetic tree by one having much fewer nodes typically does not destroy too much branching information. To demonstrate the algorithmic advantage of using trees with few internal nodes, we also propose a new algorithm for computing the rooted triplet distance that is faster than the existing algorithms when restricted to such trees.

We study the planar orthogonal drawing style within the framework of partial representation extension. Let $$(G,H,\varGamma _H)$$ be a partial orthogonal drawing, i.e., G is a graph, $$H\subseteq G$$ is a subgraph and $$\varGamma _H$$ is a planar orthogonal drawing of H. We show that the existence of an orthogonal drawing $$\varGamma _G$$ of G that extends $$\varGamma _H$$ can be tested in linear time. If such a drawing exists, then there also is one that uses O(|V(H)|) bends per edge. On the other hand, we show that it is NP-complete to find an extension that minimizes the number of bends or has a fixed number of bends per edge.

We combine two fundamental, previously studied optimization problems related to the construction of phylogenetic trees called maximum rooted triplets consistency (MAXRTC) and minimally resolved supertree (MINRS) into a new problem, which we call q-maximum rooted triplets consistency (q-MAXRTC). The input to our new problem is a set R of resolved triplets (rooted, binary phylogenetic trees with three leaves each) and the objective is to find a phylogenetic tree with exactly q internal nodes that contains the largest possible number of triplets from R. We first prove that q-MAXRTC is NP-hard even to approximate within a constant ratio for every fixed q >= 2, and then develop various polynomial-time approximation algorithms for different values of q. Next, we show experimentally that representing a phylogenetic tree by one having much fewer nodes typically does not destroy too much triplet branching information. As an extreme example, we show that allowing only nine internal nodes is still sufficient to capture on average 80% of the rooted triplets from some recently published trees, each having between 760 and 3081 internal nodes. Finally, to demonstrate the algorithmic advantage of using trees with few internal nodes, we propose a new algorithm for computing the rooted triplet distance between two phylogenetic trees over a leaf label set of size n that runs in O(q n) time, where q is the number of internal nodes in the smaller tree, and is therefore faster than the currently best algorithms for the problem (with O(n log n) time complexity [SODA 2013, ESA 2017]) whenever q = o(log n).