Summary: An edge labeling of a connected graph \(G = (V, E)\) is said to be local antimagic if it is a bijection \(f:E \rightarrow \{1,\dots,|E|\}\) such that for any pair of adjacent vertices \(x\) and \(y\), \(f^+(x) \neq f^+(y)\), where the induced vertex label \(f^+(x)= \sum f(e)\), with \(e\) ranging over all the edges incident to \(x\). The local antimagic chromatic number of \(G\), denoted by \(_{\chi_{la}}(G)\), 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 [S. Arumugam et al., Graphs Comb. 33, No. 2, 275–285 (2017; Zbl 1368.05124)].