Finding a Maximum Clique in a Grounded 1-Bend String Graph

07/12/2021 ∙ by J. Mark Keil, et al. ∙ 0

A grounded 1-bend string graph is an intersection graph of a set of polygonal lines, each with one bend, such that the lines lie above a common horizontal line ℓ and have exactly one endpoint on ℓ. We show that the problem of finding a maximum clique in a grounded 1-bend string graph is APX-hard, even for strictly y-monotone strings. For general 1-bend strings, the problem remains APX-hard even if we restrict the position of the bends and end-points to lie on at most three parallel horizontal lines. We give fast algorithms to compute a maximum clique for different subclasses of grounded segment graphs, which are formed by restricting the strings to various forms of L-shapes.

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