EPG Graphs Research Articles

B2-EPG Split

On this research we study EPG graphs, in particular, we are interested on investigate the intersection between split and B2-EPGgraph classes. The results found in the literature regarding split graphs only concern about characterization in B1-EPG, that despite bringing some graphs that belong to this class, the bend number of split graphs it is still unknown. We study the split graphs whose degree of vertices on independent set is less than 2. In this work we manipulate bipartite graphs and we present representation for some bipartite graphs in B2-EPG. We build an algorithm that create a Split B2-EPG representation, also we present another algorithm that builds a representation on a grid Qw×2y+1 for any Split graph, and other results.

Relationship among B1-EPG, VPT and EPT graphs classes

Relationship among <i>B₁</i>-EPG, VPT and EPT graphs classes

On independent set in [formula omitted]-EPG graphs

In this paper we consider the Maximum Independent Set problem (MIS) on B1-EPG graphs, that is the one-bend (B1) Edge intersection graphs of Paths on a Grid (EPG graphs). The graph class EPG was introduced in Golumbic et al. (2019) as the class of graphs whose vertices can be represented as simple paths on a rectangular grid so that two vertices are adjacent if and only if the corresponding paths share at least one edge of the underlying grid. The restricted class Bk-EPG denotes EPG-graphs where every path has at most k bends. The study of MIS on B1-EPG graphs has been initiated in Epstein et al. (2013) where authors prove that MIS is NP-complete on B1-EPG graphs, and provide a polynomial 4-approximation. In this article we study the approximability and the fixed parameter tractability of MIS on B1-EPG. We show that the class of k≥4 subdivided graphs is a subclass of B1-EPG, even if there is only one shape of path and if each path has its vertical part or its horizontal part of length at most 1. This implies that there is no PTAS for MIS (and several other classical problems) on these particular B1-EPG graphs. On the positive side, we show that if the length of the horizontal part of every path is bounded by a constant, then MIS admits a PTAS. Finally, we show that MIS is FPT in the standard parameterization on B1-EPG restricted to only three shapes of path, and W[1]-hard on B2-EPG. The status for general B1-EPG (with the four shapes) is left open.

Hardness and approximation for L-EPG and [formula omitted]-EPG graphs

The edge intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as simple paths on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. We consider the case of single-bend paths and its subclass of ⌞-shaped paths, namely, the classes known as B1-EPG and ⌞-EPG graphs, respectively.We show that fully-subdivided graphs are ⌞-EPG graphs, and later use this result in order to show that the following problems are APX-hard on ⌞-EPG graphs: Minimum Vertex Cover, Maximum Independent Set, and Maximum Weighted Independent Set, and also that Minimum Dominating Set is NP-complete on ⌞-EPG graphs. We also observe that Minimum Coloring is NP-complete already on ⌞-EPG, which follows from a proof for B1-EPG in Epstein et al. (2013). Finally, we provide efficient constant-factor-approximation algorithms for each of these problems on B1-EPG graphs.

Edge-intersection graphs of boundary-generated paths in a grid

Edge-intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as simple paths on a rectangular grid such that two vertices are adjacent in the graph if and only if the corresponding paths share at least one edge of the grid. For two boundary points p and q on two adjacent boundaries of a rectangular grid G, we call the unique single-bend path connecting p and q in G using no other boundary point of G as the path generated by (p,q). A path in G is called boundary-generated, if it is generated by some pair of points on two adjacent boundaries of G. In this article, we study the edge-intersection graphs of boundary-generated paths on a grid or ∂EPG graphs. The motivation for studying these graphs comes from problems in the context of circuit layout.We show that ∂EPG graphs can be covered by two collections of vertex-disjoint co-bipartite chain graphs. This leads us to a linear-time testable characterization of ∂EPG trees and also an almost tight upper bound on the equivalence covering number of general ∂EPG graphs. We also study the cases of two-sided ∂EPG and three-sided ∂EPG graphs, which are respectively, the subclasses of ∂EPG graphs obtained when all the boundary-vertex pairs which generate the paths are restricted to lie on at most two or three boundaries of the grid. For the former case, we give a complete characterization.

On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern [12] proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, Bk-EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B4-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that circular-arc graphs are B3-EPG, and that there exist circular-arc graphs which are not B2-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in the boundary of a rectangle of the grid), we obtain EPR (edge intersection of paths in a rectangle) representations. We may define Bk-EPR graphs, k≥0, the same way as Bk-EPG graphs. Circular-arc graphs are clearly B4-EPR graphs and we will show that there exist circular-arc graphs that are not B3-EPR graphs. We also show that normal circular-arc graphs are B2-EPR graphs and that there exist normal circular-arc graphs that are not B1-EPR graphs. Finally, we characterize B1-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs.

Edge Intersection Graphs of L-Shaped Paths in Grids

In this paper we continue the study of the edge intersection graphs of single bend paths on a rectangular grid (i.e., the edge intersection graphs where each vertex is represented by one of the following shapes: ⌞,⌜,⌟,⌝). These graphs, called B1- EPG graphs, were first introduced by Golumbic et al (2009) [Golumbic, M. C., M. Lipshteyn and M. Stern, Edge intersection graphs of single bend paths on a grid, Networks 54:3 (2009), 130–138]. We focus on the class [⌞] (the edge intersection graphs of ⌞-shapes) and show that testing for membership in [⌞] is NP-complete. We then give a characterization and polytime recognition algorithm for special subclasses of Split∩[⌞]. We also consider the natural subclasses of B1-EPG formed by the subsets of the four single bend shapes (i.e., {⌞},{⌞,⌜},{⌞,⌝},{⌞,⌜,⌝} – note: all other subsets are isomorphic to these up to 90 degree rotation). We observe the expected strict inclusions and incomparability (i.e., [⌞]⊊[⌞,⌜],[⌞,⌝]⊊[⌞,⌜,⌝]⊊B1-EPG and [⌞,⌜] is incomparable with [⌞,⌝]).

Some properties of edge intersection graphs of single-bend paths on a grid

In this paper we consider graphs G whose vertices can be represented as single-bend paths (i.e., paths with at most one turn) on a rectangular grid, such that two vertices are adjacent in G if and only if the corresponding paths share at least one edge of the grid. These graphs, called B 1 -EPG graphs, were first introduced in Golumbic et al. (2009) [13]. Here we show that the neighborhood of every vertex in a B 1 -EPG graph induces a weakly chordal graph. From this we conclude that the family F of B 1 -EPG graphs satisfies the Erdős–Hajnal property with ϵ ( F ) = 1 3 , i.e., that every B 1 -EPG graph on n vertices contains either a clique or a stable set of size at least n 1 3 . Finally we give a characterization of B 1 -EPG graphs among some subclasses of chordal graphs, namely chordal bull-free graphs, chordal claw-free graphs, chordal diamond-free graphs, and special cases of split graphs.

Edge intersection graphs of systems of paths on a grid with a bounded number of bends

We answer some of the questions raised by Golumbic, Lipshteyn and Stern and obtain some other results about edge intersection graphs of paths on a grid (EPG graphs). We show that for any d ≥ 4 , in order to represent every n vertex graph with maximum degree d as an edge intersection graph of n paths on a grid, a grid of area Θ ( n 2 ) is needed. We also show several results related to the classes B k -EPG, where B k -EPG denotes the class of graphs that have an EPG representation such that each path has at most k bends. In particular, we prove: For a fixed k and a sufficiently large n , the complete bipartite graph K m , n does not belong to B 2 m − 3 -EPG (it is known that this graph belongs to B 2 m − 2 -EPG); for any odd integer k we have B k -EPG ⫋ B k + 1 -EPG; there is no number k such that all graphs belong to B k -EPG; only 2 O ( k n log ( k n ) ) out of all the 2 n 2 labeled graphs with n vertices are in B k -EPG.