Circular-arc graphs as edge intersection graphs of paths on a grid.

LILIANA ALCÓN; FLAVIA BONOMO; GUILLERMO DURÁN; MARISA GUTIERREZ; MARÍA PÍA MAZZOLENI; BERNARD RIES; MARIO VALENCIA-PABON

Barranquilla

Congreso; V Congreso Latinoamericano de Matemáticos (CLAM) 2016; 2016

Golumbic, Lipshteyn and Stern proved that every graph can be repre-sented as the edge intersection graph of paths on a grid (EPG graph), i.e., onecan associate with each vertex of the graph a nontrivial path on a rectangulargrid such that two vertices are adjacent if and only if the corresponding pathsshare at least one edge of the grid. For a nonnegative integer k, Bk-EPGgraphs are defined as EPG graphs admitting a model in which each path hasat most k bends. Circular-arc graphs are intersection graphs of open arcs ofa 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 provethat circular-arc graphs are B3-EPG, and that there exist circular-arc graphswhich 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 boundaryof 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 sameway as Bk-EPG graphs. Circular-arc graphs are clearly B4-EPR graphs andwe 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 thatthere exist normal circular-arc graphs that are not B1-EPR graphs. Finally,we characterize B1-EPR graphs by a family of minimal forbidden inducedsubgraphs, and show that they form a subclass of normal Helly circular-arcgraphs.