The two-layer crossing minimization problem (TLCM), given a bipartite graph G with n vertices and a positive integer k, asks whether G has a two-layer drawing with at most k crossings. We consider a simple generalization of TLCM called leaf-edge-weighted TLCM (LEW-TLCM), where we allow positive weights on edges incident to leaves, and show that this problem admits a kernel with O (k) edges provided that the given graph is connected.As a straightforward consequence, LEW-TLCM (and hence TLCM) has a fixed parameter algorithm that runs in 2 O (k log k) +n O (1) time which improves on the previously best known algorithm with running time 2 O (k 3 ) n.
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