The sides $ a,b$ and the bisector of the included angle $ \gamma$ of a triangle are given. Determine necessary and sufficient conditions for such triangles to be constructible and show how to reconstruct the triangle.
1. **Given Data and Definitions:**
   - Let \( a \) and \( b \) be the sides of the triangle.
   - Let \( \gamma \) be the included angle between sides \( a \) and \( b \).
   - Let \( CV = w_c \) be the length of the angle bisector of \( \angle C \).

2. **Constructing Point \( M \):**
   - Take the point \( M \) on ray \( BC \) such that \( AC = CM = b \).

3. **Similarity of Triangles:**
   - Since \( \triangle AMB \sim \triangle WCB \), we have:
     \[
     \frac{AM}{w_c} = \frac{a + b}{a}
     \]
   - Solving for \( AM \):
     \[
     AM = \frac{w_c(a + b)}{a}
     \]

4. **Constructing Isosceles Triangle \( \triangle ACM \):**
   - The length \( AM \) is constructible.
   - Construct the isosceles triangle \( \triangle ACM \) with \( AC = CM = b \) and \( AM = \frac{w_c(a + b)}{a} \).

5. **Constructing Point \( B \):**
   - Construct point \( B \) on the ray \( MC \) such that \( CB = a \).

6. **Necessary Condition for Constructibility:**
   - For \( \triangle ABC \) to be constructible, \( \triangle ACM \) must also be constructible.
   - The triangle inequality for \( \triangle ACM \) gives:
     \[
     AC + CM > AM
     \]
   - Substituting the known lengths:
     \[
     b + b > \frac{w_c(a + b)}{a}
     \]
   - Simplifying:
     \[
     2b > \frac{w_c(a + b)}{a}
     \]
   - Multiplying both sides by \( a \):
     \[
     2ab > w_c(a + b)
     \]
   - Dividing both sides by \( a + b \):
     \[
     \frac{2ab}{a + b} > w_c
     \]

Conclusion:
The necessary and sufficient condition for the triangle \( \triangle ABC \) to be constructible is:
\[
\frac{2ab}{a + b} > w_c
\]

The final answer is \( \boxed{ \frac{2ab}{a + b} > w_c } \)