A linear time algorithm for metric dimension of cactus block graphs

An undirected graph G=(V,E)G=(V,E) has metric dimension at most k   if there is a vertex set U⊆VU⊆V such that |U|≤k|U|≤k and ∀u,v∈V∀u,v∈V, u≠vu≠v, there is a vertex w∈Uw∈U such that dG(w,u)≠dG(w,v)dG(w,u)≠dG(w,v), where dG(u,v)dG(u,v) is the distance (the length of a shortest path in an unweighted graph) between u and v. The metric dimension of G is the smallest integer k such that G has metric dimension at most k. A cactus block graph is an undirected graph whose biconnected components are either cycles or complete graphs. We present a linear time algorithm for computing the metric dimension of cactus block graphs.