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- W2888077354 abstract "Due to data compression or low resolution, nearby vertices and edges of a graph drawn in the plane may be bundled to a common node or arc. We model such a “compromised” drawing by a piecewise linear map $$varphi :Grightarrow {mathbb {R}}^2$$ . We wish to perturb $$varphi $$ by an arbitrarily small $$varepsilon >0$$ into a proper drawing (in which the vertices are distinct points, any two edges intersect in finitely many points, and no three edges have a common interior point) that minimizes the number of crossings. An $$varepsilon $$ -perturbation, for every $$varepsilon >0$$ , is given by a piecewise linear map $$psi _varepsilon :Grightarrow {mathbb {R}}^2$$ with $$Vert varphi -psi _varepsilon Vert <varepsilon $$ , where $$Vert .Vert $$ is the uniform norm (i.e., $$sup $$ norm). We present a polynomial-time solution for this optimization problem when G is a cycle and the map $$varphi $$ has no spurs (i.e., no two adjacent edges are mapped to overlapping arcs). We also show that the problem becomes NP-complete (i) when G is an arbitrary graph and $$varphi $$ has no spurs, and (ii) when $$varphi $$ may have spurs and G is a cycle or a union of disjoint paths." @default.
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- W2888077354 date "2020-06-12" @default.
- W2888077354 modified "2023-09-27" @default.
- W2888077354 title "Crossing minimization in perturbed drawings" @default.
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- W2888077354 doi "https://doi.org/10.1007/s10878-020-00586-0" @default.
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