String graphs of k-bend paths on a grid

Abstract

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.