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.