Algorithmic aspect on the minimum (weighted) doubly resolving set problem of graphs

Let G be a simple graph, where each vertex has a nonnegative weight. A vertex subset S of G is a doubly resolving set (DRS) of G if for every pair of vertices u, v in G, there exist $$x,y\in S$$x,y∈S such that $$d(x,u)-d(x,v)\ne d(y,u)-d(y,v)$$d(x,u)-d(x,v)≠d(y,u)-d(y,v). The minimum weighted doubly resolving set (MWDRS) problem is finding a doubly resolving set with minimum total weight. We establish a linear time algorithm for the MWDRS problem of all graphs in which each block is complete graph or cycle. Hence, the MWDRS problems for block graphs and cactus graphs can be solved in linear time. We also prove that k-edge-augmented tree (a tree with additional k edges) with minimum degree $$\delta (G)\ge 2$$δ(G)≥2 admits a doubly resolving set of size at most $$2k+1$$2k+1. This implies that the DRS problem on k-edge-augmented tree can be solved in $$O(n^{2k+3})$$O(n2k+3) time.