In theory Implicit (Backward) Euler should be a Runge-Kutta method with tableu given below, however I find that the standard Backward Euler formula and the Runge-Kutta one differ. $$ {\begin{array}{c|c}1&1\\\hline &1\\\end{array}} $$ Here are the two methods:
Standard Implicit Euler $y_{n+1} = y_n + h f(t_{n+1}, y_{n+1})$ where $t_{n+1} = t_n + h$.
Runge-Kutta with tableu above gives $$ \begin{align} k_1 &= f(t_0 + h, y_0 + hk_1) \\ y_1 &= y_0 + hk_1 \end{align} $$
Runge Kutta formulas
I have used the formulas in Geometric Numerical Integration by Hairer $$ \begin{align} k_i &= f\left(t_0 + c_i h, y_0 + h\sum_{i=1}^s a_{ij} k_j\right) \\ y_1 &= y_0 + h\sum_{i=1}^s b_i k_i \end{align} $$ and Implicit Euler has $c_1 = 1$, $a_{11} = 1$ and $b_1 = 1$.