Every outerplanar graph has a monotonic edge-intersection-of-paths-on-a-grid representation with at most two bends, and maximal outerplanar graphs and cacti are fully classified by forbidden induced subgraphs for bend numbers 0, 1, and 2.
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Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid
Every outerplanar graph has a monotonic edge-intersection-of-paths-on-a-grid representation with at most two bends, and maximal outerplanar graphs and cacti are fully classified by forbidden induced subgraphs for bend numbers 0, 1, and 2.