On the Recognition of Four-Directional Orthogonal Ray Graphs

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

Vertex-Weighted Matching in Two-Directional Orthogonal Ray Graphs

Let G denote an n-vertex two-directional orthogonal ray graph. A bicolored 2D representation of G requires only O(n) space, regardless of the number of edges in G. Given such a compact representation of G, and a (possibly negative) weight for each vertex, we show how to compute a maximum weight matching of G in O(n log 2 n) time. The classic problem of scheduling weighted unit tasks with release times and deadlines is a special case of this problem, and we obtain an O(n log n) time bound for this special case. As an application of our more general result, we obtain an O(n log 2 n)-time algorithm for computing the VCG outcome of a sealed-bid unit-demand auction in which each item has two associated numerical parameters (e.g., third-party “quality ” and “seller reliability ” scores) and each bid specifies the amount an agent is willing to pay for any item meeting specified lower bound constraints with respect to these two parameters

Intersection Dimension of Bipartite Graphs

We introduce a concept of intersection dimension of a graphwith respect to a graph class. This generalizes Ferrers dimension, boxicity, and poset dimension, and leads to interesting new problems. We focus in particular on bipartite graph classes defined as intersection graphs of two kinds of geometric objects. We relate well-known graph classes such as interval bigraphs, two-directional orthogonal ray graphs, chain graphs, and (unit) grid intersection graphs with respect to these dimensions. As an application of these graphtheoretic results, we show that the recognition problems for certain graph classes are NP-complete.Theory and Applications of Models of Computation, 11th Annual Conference, TAMC 2014, Chennai, India, April 11-13, 2014. Proceeding