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Keywords:
graph; degree sequence; potentially $K_m-G$-graphic sequences
Summary:
The set of all non-increasing nonnegative integer sequences $\pi =$ ($d(v_1 ),d(v_2 ), \dots , d(v_n )$) is denoted by ${\rm NS}_n$. A sequence $\pi \in {\rm NS}_n$ 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 ${\rm NS}_n$ is denoted by ${\rm GS}_n$. 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 a complete graph on $k$ vertices. Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_m$). This paper summarizes briefly some recent results on potentially $K_m-G$-graphic sequences and give a useful classification for determining $\sigma (H,n)$.
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