Given two graphs \(H_1\) and \(H_2\) , a graph is \((H_1,H_2)\) -free if it contains no induced subgraph isomorphic to \(H_1\) nor \(H_2\) . A graph G is k-vertex-critical if every proper induced subgraph of G has chromatic number less than k, but G has chromatic number k. The study of k-vertex-critical graphs for specific graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there exists a polynomial-time certifying algorithm to decide the k-colorability of a graph in the class. In this paper, we show that: (1) for \(k \ge 1\) , there are finitely many k-vertex-critical \((P_5,K_{1,4}+P_1)\) -free graphs; (2) for \(s \ge 1\) , there are finitely many 5-vertex-critical \((P_5,K_{1,s}+P_1)\) -free graphs; (3) for \(k \ge 1\) , there are finitely many k-vertex-critical \((P_5,\overline{K_3+2P_1})\) -free graphs. Moreover, we characterize all 5-vertex-critical \((P_5,H)\) -free graphs where \(H \in \{K_{1,3}+P_1,K_{1,4}+P_1,\overline{K_3+2P_1}\}\) using an exhaustive graph generation algorithm.