  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 $H_2$. Let $P_t$ be the path on $t$
vertices. A dart is the graph obtained from a diamond by adding a new vertex
and making it adjacent to exactly one vertex with degree 3 in the diamond.
  In this paper, we show that there are finitely many $k$-vertex-critical
$(P_5,dart)$-free graphs for $k \ge 1$ To prove these results, we use induction
on $k$ and perform a careful structural analysis via Strong Perfect Graph
Theorem combined with the pigeonhole principle based on the properties of
vertex-critical graphs. Moreover, for $k \in \{5, 6, 7\}$ we characterize all
$k$-vertex-critical $(P_5,dart)$-free graphs using a computer generation
algorithm. Our results imply the existence of a polynomial-time certifying
algorithm to decide the $k$-colorability of $(P_5,dart)$-free graphs for $k \ge
1$ where the certificate is either a $k$-coloring or a $(k+1)$-vertex-critical
induced subgraph.
