The set of all non-increasing nonnegative integer sequences $\pi =$ ($d\left(v_1\right),d\left(v_2\right),\cdots ,d\left(v_n\right)$) is denoted by ${\mathrm{NS}}_n$. A sequence $\pi \in {\mathrm{NS}}_n$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices, and such a graph $G$ is called a realization of $\pi$. The set of all graphic sequences in ${\mathrm{NS}}_n$ is denoted by ${\mathrm{GS}}_n$. A graphical sequence $\pi$ is potentially $H$-graphical if there is a realization of $\pi$ containing $H$ as a subgraph, while $\pi$ is forcibly $H$-graphical if every realization of $\pi$ contains $H$ as a subgraph. Let $K_k$ denote...

Let $r\ge 3$, $n\ge r$ and $\pi =(d_1,d_2,...,d_n)$ be a non-increasing sequence of nonnegative integers. If $\pi$ has a realization $G$ with vertex set $V\left(G\right)=\{v_1,v_2,...,v_n\}$ such that $d_G\left(v_i\right)=d_i$ for $i=1,2,...,n$ and $v_1{v}_2\cdots v_r{v}_1$ is a cycle of length $r$ in $G$, then $\pi$ is said to be potentially $C_r^{\text{'}\text{'}}$-graphic. In this paper, we give a characterization for $\pi$ to be potentially $C_r^{\text{'}\text{'}}$-graphic.

Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E\left(H\right)$ of $H$ where $H$ is a subgraph of $K_m$. In this paper, we characterize the potentially $K_5-P_4$ and $K_5-Y_4$-graphic sequences where $Y_4$ is a tree on 5 vertices and 3 leaves.