The following is the Hasse diagram of the poset $$\left[ {\left\{ {a,b,c,d,e} \right\}, \prec } \right]$$

The poset is:

The determination of the matrix given below is $$$\left[ {\matrix{ 0 & 1 & 0 & 2 \cr { - 1} & 1 & 1 & 3 \cr 0 & 0 & 0 & 1 \cr 1 & { - 2} & 0 & 1 \cr } } \right]$$$

Let $$G$$ be a simple connected planar graph with 13 vertices and 19 edges. Then, the number of faces in the planar embedding of the graph is:

Let $$G$$ be the simple graph with 20 vertices and 100 edges. The size of the minimum vertex cover of $$G$$ is 8. Then, the size of the maximum independent set of $$G$$ is: