Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid
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A graph G is called an edge intersection graph of paths on a grid if there is a grid and there is a set of paths on this grid, such that the vertices of G correspond to the paths and two vertices of G are adjacent if and only if the corresponding paths share a grid edge. Such a representation is called an EPG representation of G. B_k is the class of graphs for which there exists an EPG representation where every path has at most k bends. The bend number b(G) of a graph G is the smallest natural number k for which G belongs to B_k. B_k^m is the subclass of B_k containing all graphs for which there exists an EPG representation where every path has at most k bends and is monotonic, i.e. it is ascending in both columns and rows. The monotonic bend number b^m(G) of a graph G is the smallest natural number k for which G belongs to B_k^m. Edge intersection graphs of paths on a grid were introduced by Golumbic, Lipshteyn and Stern in 2009 and a lot of research has been done on them since then. In this paper we deal with the monotonic bend number of outerplanar graphs. We show that b^m(G)≤ 2 holds for every outerplanar graph G. Moreover, we characterize in terms of forbidden subgraphs the maximal outerplanar graphs and the cacti with (monotonic) bend number equal to 0, 1 and 2. As a consequence we show that for any maximal outerplanar graph and any cactus a (monotonic) EPG representation with the smallest possible number of bends can be constructed in a time which is polynomial in the number of vertices of the graph.
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