We introduce and study the OrthoSEFE-k problem: Given k planar graphs each with maximum degree 4 and the same vertex set, do they admit an OrthoSEFE, that is, is there an assignment of the vertices to grid points and of the edges to paths on the grid such that the same edges in distinct graphs are assigned the same path and such that the assignment induces a planar orthogonal drawing of each of the k graphs? We show that the problem is NP-complete for k â¥ 3 even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for k â¥ 2 even if the shared graph consists of a cycle and of isolated vertices. Whereas the problem is polynomial-time solvable for k = 2 when the union graph has maximum degree five and the shared graph is biconnected. Further, when the shared graph is biconnected and has sunflower intersection, we show that every positive instance has an OrthoSEFE with at most three bends per edge.

Angelini, P., Chaplick, S., Cornelsen, S., Da Lozzo, G., Di Battista, G., Eades, P., et al. (2016). Simultaneous orthogonal planarity. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (pp.532-545). Springer Verlag [10.1007/978-3-319-50106-2_41].

### Simultaneous orthogonal planarity

#### Abstract

We introduce and study the OrthoSEFE-k problem: Given k planar graphs each with maximum degree 4 and the same vertex set, do they admit an OrthoSEFE, that is, is there an assignment of the vertices to grid points and of the edges to paths on the grid such that the same edges in distinct graphs are assigned the same path and such that the assignment induces a planar orthogonal drawing of each of the k graphs? We show that the problem is NP-complete for k â¥ 3 even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for k â¥ 2 even if the shared graph consists of a cycle and of isolated vertices. Whereas the problem is polynomial-time solvable for k = 2 when the union graph has maximum degree five and the shared graph is biconnected. Further, when the shared graph is biconnected and has sunflower intersection, we show that every positive instance has an OrthoSEFE with at most three bends per edge.
##### Scheda breve Scheda completa Scheda completa (DC)
9783319501055
Angelini, P., Chaplick, S., Cornelsen, S., Da Lozzo, G., Di Battista, G., Eades, P., et al. (2016). Simultaneous orthogonal planarity. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (pp.532-545). Springer Verlag [10.1007/978-3-319-50106-2_41].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11590/325738`
• ND
• 7
• 4