Graphical sequences of some family of induced subgraphs

The subdivision graph $s(g)$ of a graph $g$ is the graph obtained by inserting a new vertex into every edge of $g$. the $s_{vertex}$ or $s_{ver}$ join of the graph $g_{1}$ with the graph $g_{2}$, denoted by $g_{1}dot{vee}g_{2}$, is obtained from $s(g_{1})$ and $g_{2}$ by joining all vertices of $g_{1}$ with all vertices of $g_{2}$. the $s_{edge}$ or $s_{ed}$ join of $g_{1}$ and $g_{2}$, denoted by $g_{1}bar{vee}g_{2}$, is obtained from $s(g_{1})$ and $g_{2}$ by joining all vertices of $s(g_{1})$ corresponding to the edges of $g_{1}$ with all vertices of $g_{2}$. in this paper, we obtain graphical sequences of the family of induced subgraphs of $s_{j} = g_{1}vee g_{2}$, $s_{ver} = g_{1}dot{vee}g_{2}$ and $s_{ed} = g_{1}bar{vee}g_{2}$. also we prove that the graphic sequence of $s_{ed}$ is potentially $k_{4}-e$-graphical.