Abstract:
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We present a fixed parameter algorithm that constructively solves the
k-dominating set problem on graphs excluding one of the K_{5} or
K_3,3 as a minor in time O(3^{6sqrt{34 k}}n^{O(1)}). In fact,
we present our algorithm for any H-minor-free graph where H is
a single-crossing graph (can be drawn on the plane with at most one
crossing) and obtain the algorithm for K_{3,3} (K_{5})-minor-free
graphs as a special case. As a consequence, we extend our results to
several other problems such as vertex cover, edge dominating set,
independent set, clique-transversal set, kernels in digraphs,
feedback vertex set and a series of vertex removal problems.
Our work generalizes and extends the recent result of exponential speedup
in designing fixed-parameter algorithms on planar
graphs due to Alber et al. to other (non-planar) classes of graphs. |