An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a bijection f:E →{1,... ,|E|} such that for any pair of adjacent vertices x and y, f⁺(x)≠ f⁺(y), where the induced vertex label f⁺(x)= ∑ f(e), with e ranging over all the edges incident to x.  The local antimagic chromatic number of G, denoted by X_la(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pendants is obtained. The exact value of the local antimagic chromatic number of many families of graphs with cut-vertices (possibly given by pendant edges) are also determined. Consequently, we partially answered Problem 3.1 in [Local antimagic vertex coloring of a graph, Graphs and Combin., 33, (2017),  275--285].