Algorithm on rainbow connection for maximal outerplanar graphs

In this paper, we consider rainbow connection number of maximal outerplanar graphs (MOPs) on algorithmic aspect. For the (MOP) G, we give sufficient conditions to guarantee that r c ( G ) = d i a m ( G ) . Moreover, we produce the graph with given diameter D and give their rainbow coloring in linear time. X. Deng et al. 4 give a polynomial time algorithm to compute the rainbow connection number of MOPs by the Maximal fan partition method, but only obtain a compact upper bound. J. Lauri 11 proved that, for chordal outerplanar graphs given an edge-coloring, to verify whether it is rainbow connected is NP-complete under the coloring, it is so for MOPs. Therefore we construct Central-cut-spine of MOP G, by which we design an algorithm to give a rainbow edge coloring with at most 2 r a d ( G ) + 2 + c , 0 ź c ź r a d ( G ) - 2 colors in polynomial time.