Strong rainbow disconnection in graphs

Abstract

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