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We analyse properties to show the strict containment between some natural classes of geometric intersection graphs. For example, we show that, for each integer k ≥ 1 , the class of intersection graphs of segments with k different lengths is a strict subclass of the class of intersection graphs of segments with k + 1 different lengths, and the class of outer-segment graphs is a strict subclass of the class of outer-string graphs. [ABSTRACT FROM AUTHOR]

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 -VPG graphs, . 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 -VPG graphs. Chordal -VPG graphs are shown to be exactly Strongly Chordal -VPG graphs. We prove the strict containment of -VPG and circle graphs into -VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are -VPG graphs. In the case of -VPG graphs, we observe that a set of horizontal and vertical segments have strong Helly number 2. We show that the coloring problem for -VPG graphs, for , is NP-complete and give a 2-approximation algorithm for coloring -VPG graphs. Furthermore, we prove that triangle-free -VPG graphs are 4-colorable, and this is best possible. [Copyright &y& Elsevier]

Abstract: We show that for every n there are two simple curves on the torus intersecting at least n times without the two curves folding or spiralling with respect to each other. [Copyright &y& Elsevier]