Stick graphs: examples and counter-examples
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Stick graphs are the intersection graphs of vertical and horizontal segments in the plane whose bottom and respectively left endpoints belong to a line with negative slope. Such a representation is called a Stick representation of the graph. Very few results exist on Stick graphs: only trivial classes of Stick graphs have been identified; recognizing Stick graphs is an open problem; and even building examples of graphs that are not Stick graphs is quite tricky. In this paper, we first propose structural results on Stick graphs, connecting the neighborhoods of two vertices to the order of these two vertices in a possible Stick representation. We deduce local obstructions for the existence of a Stick representation, which we use for the construction of graphs that are not Stick graphs. We next disapprove the idea, suggested by these examples and the other examples from the literature, that a graph which is not a Stick graph necessarily contains a hole. Finally, we show that bipartite complements of circular-arc graphs and of circle graphs are subclasses of Stick graphs.
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