Grid Intersection Graphs Research Articles

Recognizing geometric intersection graphs stabbed by a line

In this paper, we determine the computational complexity of recognizing two graph classes, groundedL-graphs and stabbable grid intersection graphs. An L-shape is made by joining the bottom end-point of a vertical (|) segment to the left end-point of a horizontal (−) segment. The top end-point of the vertical segment is known as the anchor of the L-shape. Grounded L-graphs are the intersection graphs of L-shapes such that all the L-shapes' anchors lie on the same horizontal line. We show that recognizing grounded L-graphs is NP-complete. This answers an open question asked by Jelínek & Töpfer (Electron. J. Comb., 2019).Grid intersection graphs are the intersection graphs of axis-parallel line segments in which two vertical (similarly, two horizontal) segments cannot intersect. We say that a (not necessarily axis-parallel) straight line ℓ stabs a segment s, if s intersects ℓ. A graph G is a stabbable grid intersection graph (▪) if there is a grid intersection representation of G in which the same line stabs all its segments. We show that recognizing ▪ graphs is NP-complete, even on a restricted class of graphs. This answers an open question asked by Chaplick et al. (Order, 2018).

Grid Intersection Graphs and Order Dimension

We study subclasses of grid intersection graphs from the perspective of order dimension. We show that partial orders of height two whose comparability graph is a grid intersection graph have order dimension at most four. Starting from this observation we provide a comprehensive study of classes of graphs between grid intersection graphs and bipartite permutation graphs and the containment relation on these classes. Order dimension plays a role in many arguments.

Ferrers dimension of grid intersection graphs

We investigate the Ferrers dimension of classes of grid intersection graphs and show properties and characterizations. In particular, we show that (1) the grid intersection graphs form a proper subclass of the class of bipartite graphs of Ferrers dimension 4, (2) segment-ray graphs have a forbidden submatrix characterization, and (3) a bipartite graph is a unit grid intersection graph if and only if it is the intersection of two bipartite permutation graphs.

The Order Dimension of Planar Maps Revisited

Schnyder characterized planar graphs in terms of order dimension. The structures used for the proof have found many applications. Researchers also found several extensions of the seminal result. A particularly far-reaching extension is the Brightwell--Trotter theorem about planar maps. It states that the order dimension of the incidence poset $\mathbf{P}_{\mathbf{M}}$ of vertices, edges, and faces of a planar map $\mathbf{M}$ has dimension at most 4. The original proof generalizes the machinery of Schnyder paths and Schnyder regions. In this short paper we use a simple result about the order dimension of grid intersection graphs to show a slightly stronger result: $\dim(\mathsf{split}(\mathbf{P}_{\mathbf{M}})) \leq 4$. Here, $\mathsf{split}(P)$ refers to a particular order of height two associated with $P$. The Brightwell--Trotter theorem follows because $\dim(\mathsf{split}(P)) \geq \dim(P)$ holds for every $P$. This may be the first result in the area that is obtained without using the tools introduced by Schnyder.

On Minimum Feedback Vertex Sets in Bipartite Graphs and Degree-Constraint Graphs

SUMMARY We consider the minimum feedback vertex set problem for some bipartite graphs and degree-constrained graphs. We show that the problem is linear time solvable for bipartite permutation graphs and NP-hard for grid intersection graphs. We also show that the problem is solvable in O(n 2 log 6 n) time for n-vertex graphs with maximum degree at

String graphs of k-bend paths on a grid

Abstract We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the B k -VPG graphs, k ⩾ 0 . We present a complete hierarchy of VPG graphs relating them to other known families of graphs. String graphs are equivalent to VPG graphs. The grid intersection graphs [S. Bellantoni, I. Ben-Arroyo Hartman, T. Przytycka, S. Whitesides, Grid intersection graphs and boxicity, Discrete Math. 114, (1993), 41–49; I. Ben-Arroyo Hartman, I. Newman, R. Ziv, On grid intersection graphs, Discrete Math. 87(1), (1991), 41–52] are shown to be equivalent to the bipartite B 0 -VPG graphs. Chordal B 0 -VPG graphs are shown to be exactly Strongly Chordal B 0 -VPG graphs. We prove the strict containment of B 0 -VPG and circle graphs into B 1 -VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are B 3 -VPG graphs. In the case of B 0 -VPG graphs, we observe that a set of horizontal and vertical segments have strong Helly number 2. We show that the coloring problem for B k -VPG graphs, for k ⩾ 0 , is NP-complete and give a 2-approximation algorithm for coloring B 0 -VPG graphs. Furthermore, we prove that triangle-free B 0 -VPG graphs are 4-colorable, and this is best possible.

On orthogonal ray graphs

An orthogonal ray graph is an intersection graph of horizontal and vertical rays (half-lines) in the x y -plane. An orthogonal ray graph is a 2-directional orthogonal ray graph if all the horizontal rays extend in the positive x -direction and all the vertical rays extend in the positive y -direction. We first show that the class of orthogonal ray graphs is a proper subset of the class of unit grid intersection graphs. We next provide several characterizations of 2-directional orthogonal ray graphs. Our first characterization is based on forbidden submatrices. A characterization in terms of a vertex ordering follows immediately. Next, we show that 2-directional orthogonal ray graphs are exactly those bipartite graphs whose complements are circular arc graphs. This characterization implies polynomial-time recognition and isomorphism algorithms for 2-directional orthogonal ray graphs. It also leads to a characterization of 2-directional orthogonal ray graphs by a list of forbidden induced subgraphs. We also show a characterization of 2-directional orthogonal ray trees, which implies a linear-time algorithm to recognize such trees. Our results settle an open question of deciding whether a ( 0 , 1 ) -matrix can be permuted to avoid the submatrices [ 1 0 0 1 ] and [ 1 0 1 1 ] .

Interval bigraphs are unit grid intersection graphs

We give a short, elementary proof that the interval bigraphs are a strict subclass of the unit grid intersection graphs.

Relationships between the class of unit grid intersection graphs and other classes of bipartite graphs

We show that the class of unit grid intersection graphs properly includes both of the classes of interval bigraphs and of P 6 -free chordal bipartite graphs. We also demonstrate that the classes of unit grid intersection graphs and of chordal bipartite graphs are incomparable.

Optimal grid representations

AbstractA graph G is a grid intersection graph if G is the intersection graph of ℋ︁ ∪ ℐ, where ℋ︁ and ℐ are, respectively, finite families of horizontal and vertical linear segments in the plane such that no two parallel segments intersect. (This definition implies that every grid intersection graph is bipartite.) The family ℋ︁ ∪ ℐ is a representation of G. As a consequence of a characterization of grid intersection graphs by Kratochvíl, we observe that when a bipartite graph G = (U ∪ W, E) with minimum degree at least two is a grid intersection graph, then there exists a normalized representation of G on the (r × s)‐grid for r = |U| and s = |W|, that is, a representation in which all end points of segments have integer‐valued coordinates belonging to {(x, y) ∈ N × N | 1 ≤ y ≤ r, 1 ≤ x ≤ s} and the representative segment of each vertex lies on a distinct horizontal or vertical line. A natural problem, with potential applications to circuit layout, is the following: among all the possible normalized representations of G, find a representation ℛ such that the sum of the lengths of the segments in ℛ is minimum. In this work we introduce this problem and present a mixed integer programming formulation to solve it. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(3), 187–193 2004

A special planar satisfiability problem and a consequence of its NP-completeness

We introduce a weaker but still NP-complete satisfiability problem to prove NP-completeness of recognizing several classes of intersection graphs of geometric objects in the plane, including grid intersection graphs and graphs of boxicity two.

Intersection Dimensions of Graph Classes

The intersection dimension of a graphG with respect to a classA of graphs is the minimumk such thatG is the intersection of at mostk graphs on vertex setV(G) each of which belongs toA. We consider the question when the intersection dimension of a certain family of graphs is bounded or unbounded. Our main results are (1) ifA is hereditary, i.e., closed on induced subgraphs, then the intersection dimension of all graphs with respect toA is unbounded, and (2) the intersection dimension of planar graphs with respect to the class of permutation graphs is bounded. We also give a simple argument based on [Ben-Arroyo Hartman, I., Newman, I., Ziv, R.:On grid intersection graphs, Discrete Math.87 (1991) 41-52] why the boxicity (i.e., the intersection dimension with respect to the class of interval graphs) of planar graphs is bounded. Further we study the relationships between intersection dimensions with respect to different classes of graphs.

Grid intersection graphs and boxicity

A graph has boxicity k if k is the smallest integer such that G is an intersection graph of k-dimensional boxes in a k-dimensional space (where the sides of the boxes are parallel to the coordinate axis). A graph has grid dimension k if k is the smallest integer such that G is an intersection graph of k-dimensional boxes (parallel to the coordinate axis) in a ( k + 1)-dimensional space. We prove that all bipartite graphs with boxicity two, have grid dimensions one, that is, they can be represented as intersection graphs of horizontal and vertical intervals in the plane. We also introduce some inequalities for the grid dimension of a graph, and discuss extremal graphs with large grid dimensions.

On grid intersection graphs

A bipartite graph G = ( X, Y; E) has a grid representation if X and Y correspond to sets of horizontal and vertical segments in the plane, respectively, such that ( x i , y j ) ∈ E if and only if segments x i and y j intersect. We prove that all planar bipartite graphs have a grid representation, and exhibit some infinite families of graphs with no grid representation—among them the point line incidence graph of projective planes.