In this article we prove that the minimum-degree greedy algorithm, with adversarial tie-breaking, is a ( 2 / 3 ) -approximation for the Maximum Independent Set problem on interval graphs. We show that this is tight, even on unit interval graphs of maximum degree 3. We show that on chordal graphs, the greedy algorithm is a ( 1 / 2 ) -approximation and that this is again tight. These results contrast with the known (tight) approximation ratio of 3 Δ + 2 of the greedy algorithm for general graphs of maximum degree Δ .

Semi-directed networks are partially directed graphs that model evolution where the directed edges represent reticulate evolutionary events. We present an algorithm that reconstructs binary n -leaf semi-directed level-1 networks in O ( n 2 ) time from its quarnets (4-leaf subnetworks). Our method assumes we have direct access to all quarnets, yet uses only an asymptotically optimal number of O ( n log ⁡ n ) quarnets. When the network is assumed to contain no triangles, our method instead relies only on four-cycle quarnets and the splits of the other quarnets. A variant of our algorithm works with quartets rather than quarnets and we show that it reconstructs most of a semi-directed level-1 network from an asymptotically optimal O ( n log ⁡ n ) of the quartets it displays. Additionally, we provide an O ( n 3 ) time algorithm that reconstructs the tree-of-blobs of any binary n -leaf semi-directed network with unbounded level from O ( n 3 ) splits of its quarnets.