Cookies

We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.


Durham Research Online
You are in:

On the recognition of four-directional orthogonal ray graphs.

Felsner, S. and Mertzios, G.B. and Mustata, I. (2013) 'On the recognition of four-directional orthogonal ray graphs.', in Mathematical foundations of computer science 2013 : 38th international symposium, MFCS 2013, Klosterneuburg, Austria, August 26-30, 2013. Proceedings. Berlin, Heidelberg: Springer, pp. 373-384. Lecture notes in computer science. (8087).

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 TeX with both directed and undirected edges, such that G has a 4-DORG representation if and only if TeX 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.

Item Type:Book chapter
Full text:(AM) Accepted Manuscript
Download PDF
(390Kb)
Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1007/978-3-642-40313-2_34
Publisher statement:The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-40313-2_34
Date accepted:No date available
Date deposited:17 September 2014
Date of first online publication:2013
Date first made open access:No date available

Save or Share this output

Export:
Export
Look up in GoogleScholar