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

Golumbic, Lipshteyn and Stern proved that every graph can be represented as the edge intersection graph of paths on a grid, i.e., one can associate to each vertex of E-mail addresses: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]. This work was partially supported by MathAmSud Project 13MATH-07 (Argentina–Brazil– Chile–France), UBACyT Grant 20020130100808BA, CONICET PIP 122-01001-00310, 112-200901-00178 and 112-201201-00450CO and ANPCyT PICT 2010-1970 and 2012-1324 (Argentina), FONDECyT Grant 1140787 and Millennium Science Institute “Complex Engineering Systems” (Chile). the graph a nontrivial path on a 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 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 B4-EPG, by embedding the circle into a rectangle of the grid. In this paper we prove that every circular-arc graph is B3-EPG, but if we restrict ourselves to rectangular representations there exist some graphs that require paths with four bends. We also show that normal circular-arc graphs admit rectangular representations with at most two bends per path. Moreover, we characterize graphs admitting a rectangular representation with at most one bend per path by forbidden induced subgraphs, and we show that they are a subclass of normal Helly circular-arc graphs.