Strong rainbow disconnection in graphs

Abstract Let G be a nontrivial edge-colored connected graph. A rainbow edge-cut is an R of G, and all edges have different colors in G. For two vertices u v u-v-edge-cut separating them. An graph called strong disconnected if for every distinct there exists both minimum such edge-coloring disconnection coloring (srd-coloring short) the number (srd-number denoted by srd(G), required to make disconnected. In this paper, we first characterize graphs with m satisfing srd(G) = k each ∈ {1, 2, m}, respectively, also show that srd-number equal maximum blocks Secondly, study srd-numbers complete k-partite graphs, k-edge-connected k-regular grid graphs. Finally, prove computing NP-hard. particular, it NP-complete decide 3 cubic We following problem NP-complete: given (with unbounded colors) check whether makes