Abstract
Orthogonal ray graphs are the intersection graphs of horizontal and vertical rays (i.e. half-lines) in the plane. If the rays can have any possible orientation (left/right/up/down) then the graph is a 4-directional orthogonal ray graph (4-DORG). Otherwise, if all rays are only pointing into the positive x and y directions, the intersection graph is a 2-DORG. Similarly, for 3-DORGs, the horizontal rays can have any direction but the vertical ones can only have the positive direction. The recognition problem of 2-DORGs, which are a nice subclass of bipartite comparability graphs, is known to be polynomial, while the recognition problems for 3-DORGs and 4-DORGs are open. Recently it has been shown that the recognition of unit grid intersection graphs, a superclass of 4-DORGs, is NP-complete. In this paper we prove that the recognition problem of 4-DORGs is polynomial, given a partition {L,R,U,D} of the vertices of G (which corresponds to the four possible ray directions). For the proof, given the graph G, we first construct two cliques G 1,G 2 with both directed and undirected edges. Then we successively augment these two graphs, constructing eventually a graph \(\widetilde{G}\) with both directed and undirected edges, such that G has a 4-DORG representation if and only if \(\widetilde{G}\) has a transitive orientation respecting its directed edges. As a crucial tool for our analysis we introduce the notion of an S-orientation of a graph, which extends the notion of a transitive orientation. We expect that our proof ideas will be useful also in other situations. Using an independent approach we show that, given a permutation π of the vertices of U (π is the order of y-coordinates of ray endpoints for U), while the partition {L,R} of V ∖ U is not given, we can still efficiently check whether G has a 3-DORG representation.
This work was partially supported by (i) the DFG ESF EuroGIGA projects COMPOSE and GraDR, (ii) the EPSRC Grant EP/K022660/1, (iii) the EPSRC Grant EP/G043434/1, and (iv) the Berlin Mathematical School.
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References
Cabello, S., Cardinal, J., Langerman, S.: The clique problem in ray intersection graphs. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 241–252. Springer, Heidelberg (2012)
Chalopin, J., Gonçalves, D.: Every planar graph is the intersection graph of segments in the plane: extended abstract. In: Proc STOC 2009, pp. 631–638 (2009)
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9(3), 251–280 (1990)
Estivill-Castro, V., Noy, M., Urrutia, J.: On the chromatic number of tree graphs. Discrete Mathematics 223(1-3), 363–366 (2000)
Feder, T., Hell, P., Huang, J.: List homomorphisms and circular arc graphs. Combinatorica 19, 487–505 (1999)
Felsner, S.: 3-interval irreducible partially ordered sets. Order 11, 12–15 (1994)
Felsner, S., Habib, M., Möhring, R.H.: On the interplay of interval dimension and dimension. SIAM Journal on Discrete Mathematics 7, 32–40 (1994)
Habib, M., Paul, C., Viennot, L.: Partition refinement techniques: An interesting algorithmic tool kit. Int. J. Found. Comput. Sci. 10(2), 147–170 (1999)
Hartman, I.-A., Newman, I., Ziv, R.: On grid intersection graphs. Discrete Mathematics 87(1), 41–52 (1991)
Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Applied Mathematics 52(3), 233–252 (1994)
Kratochvíl, J., Matoušek, J.: NP-hardness results for intersection graphs. Commentationes Mathematicae Universitatis Carolinae 30(4), 761–773 (1989)
Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Comb. Theory, Ser. B 62(2), 289–315 (1994)
McConnell, R.: Linear-time recognition of circular-arc graphs. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pp. 386–394 (2001)
Mustaţă, I., Pergel, M.: Unit grid intersection graphs: recognition and properties (in preparation, 2013)
Otachi, Y., Okamoto, Y., Yamazaki, K.: Relationships between the class of unit grid intersection graphs and other classes of bipartite graphs. Discrete Applied Mathematics 155(17), 2383–2390 (2007)
Rao, W., Orailoglu, A., Karri, R.: Logic map** in crossbar-based nanoarchitectures. IEEE Design & Test of Computers 26(1), 68–77 (2009)
Richerby, D.: Interval bigraphs are unit grid intersection graphs. Discrete Mathematics, 1718–1719 (2009)
Shrestha, A., Tayu, S., Ueno, S.: Orthogonal ray graphs and nano-PLA design. In: Proc. IEEE ISCAS 2009, pp. 2930–2933 (2009)
Shrestha, A.M.S., Tayu, S., Ueno, S.: On orthogonal ray graphs. Discrete Appl. Math. 158(15), 1650–1659 (2010)
Soto, J.A., Telha, C.: Jump number of two-directional orthogonal ray graphs. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 389–403. Springer, Heidelberg (2011)
Trotter, W.: Combinatorics and Partially Ordered Sets. The Johns Hopkins University Press (1992)
Uehara, R.: Simple geometrical intersection graphs. In: Nakano, S.-I., Rahman, M. S. (eds.) WALCOM 2008. LNCS, vol. 4921, pp. 25–33. Springer, Heidelberg (2008)
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Felsner, S., Mertzios, G.B., Musta, I. (2013). On the Recognition of Four-Directional Orthogonal Ray Graphs. In: Chatterjee, K., Sgall, J. (eds) Mathematical Foundations of Computer Science 2013. MFCS 2013. Lecture Notes in Computer Science, vol 8087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40313-2_34
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DOI: https://doi.org/10.1007/978-3-642-40313-2_34
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