to check if the complement of the graph is bipartite or not 

Is the complement of the above two  graphs bipartite  and should the complement of the graph include  vertex 2 ?An answer explaining how the complement is bipartite or not is needed for both graphs ?
 A: The complement of the first graph is a triangle ($K_3$) plus one isolated vertex; the triangle is an odd cycle in this graph, so the graph cannot be bipartite. Alternatively, you can simply observe that no matter how you divide the four vertices into two sets, one of the sets will have to contain at least two of the vertices $1,3$, and $4$, and there is an edge between any two of these vertices.
The complement of the second graph looks like this:
        1----3    2

This is bipartite in either of two ways: we can split the vertices into the sets $\{1,2\}$ and $\{3\}$, or we can split them into the sets $\{1\}$ and $\{2,3\}$. With either of these partitions there is no edge whose ends are both in the same part, so either of them shows that the graph is bipartite. Since the only edge is between $1$ and $3$, we just have to be sure that $1$ and $3$ end up in different parts; we can put the isolated vertex $2$ in either part. (This is true in general: if a graph is bipartite, any isolated vertices can go into either part.)
A: The complement of the graph $G=(V,E)$ is defined to be the graph $\overline{G} = (V, V^{\{2\}}-E)$, where $V^{\{2\}}$ denotes the set of all $2$-subsets of $V$.  Note that the vertex set of the complement graph is the same as that of the given graph, and the edge set of the complement graph contains those edges of the complete graph on $V$ which are not present in $G$.  Thus, the union of the edge set of $G$ and the edge set of $\overline{G}$ is the edge set of the complete graph.  In your second example, the complement graph also has $3$ vertices.  In the given graph, vertex $2$ is adjacent to all other vertices, and so in the complement graph vertex $2$ will not have any neighbors.  
