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

Abstract

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, B k -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 B 4 -EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that every circular-arc graph is B 3 -EPG, and that there exist circular-arc graphs which are not B 2 -EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in a rectangle E-mail addresses: of the grid), we obtain EPR (edge intersection of path in a rectangle) representations. We may define B k -EPR graphs, k ≥ 0, the same way as B k -EPG graphs. Circulararc graphs are clearly B 4 -EPR graphs and we will show that there exist circular-arc graphs that are not B 3 -EPR graphs. We also show that normal circular-arc graphs are B 2 -EPR graphs and that there exist normal circular-arc graphs that are not B 1 -EPR graphs. Finally, we characterize B 1 -EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs.

Proper circular arc graphs as intersection graphs of pathson a grid

“…In this paper, we present characterisations by (infinite) families of minimal forbidden induced subgraphs for B 1 -EPG ∩ PCA and B 1 -EPR ∩ PCA. This is a first step towards finding a characterisation of the minimal graphs in (CA ∩ B 2 -EPG) \ (CA ∩ B 1 -EPG), a question left open in [1].…”

“…Conversely, in [11] the authors proved that PCA ∩ {W 4 , S 3 }-free = PHCA. The result then follows from the fact that PHCA ∩ {C k 4k−1 , k ≥ 2}-free ⊂ NHCA ∩ {C k 4k−1 , k ≥ 2}-free = B 1 -EPR (see [1]).…”

“…In this paper, we are interested in a subclass of circular arc graphs (CA for short), namely proper circular arc graphs. In [1], the authors showed that CA graphs are B 3 -EPG and further proved that normal circular arc graphs have bend number equal to 2, a result from which we can easily deduce that the bend number of proper circular arc graphs is 2 (see Section 2). They also considered additional constraints on the EPG representations by demanding that the union of the paths lies on the boundary of a rectangle of the grid (EPR graphs).…”

“…A graph G is said to be an edge intersection graph of paths on a rectangle (EPR graph) if there exists a set of paths P on a rectangle R of the grid in one-to-one correspondance with the vertex set of G, where two vertices are adjacent in G if and only if their corresponding paths share at least one grid-edge; (G, R, P) is then referred to as an EPR representation of G. For k ≥ 0, we denote by B k -EPR the class of graphs for which there exists an EPR representation where every path has at most k bends. The authors in [1] proved that NCA graphs have a bend number of 2 and presented an infinite family of NCA graphs, namely {C k 4k−1 , k ≥ 2}, which are not B 1 -EPG. Since any C k 4k−1 , k ≥ 2 is in fact a PCA graph, we deduce the following corollary from the fact that PCA ⊂ NCA.…”

“…90 degrees turns at a grid-point) that a path may have; for k ≥ 0, the class B k -EPG consists of those EPG graphs admitting a representation in which each path has at most k bends. Since their introduction, B k -EPG graphs have been extensively studied from several points of view (see for instance [1,2,3,5,6,7,8,9,14,15]). One major interest is the so-called bend number ; for a graph class G, the bend number of G is the minimum integer k ≥ 0 such that every graph G ∈ G is a B k -EPG graph.…”