论文标题
图形中半统治的注释
A note on semitotal domination in graphs
论文作者
论文摘要
$ g $中的$ s $顶点是$ g $的半统治集，如果它是$ g $的$ g $，并且$ s $中的每个顶点都在另一个$ s $的另一个顶点的距离$ 2 $之内。 \ emph {半统治数}，$γ_{t2}（g）$，是$ g $的半统治集的最小基数。图$ g $，$ msd_ {γ_{γ_{t2}}}（g）$的\ emph {emph {emph {emph {emph {emph {emph {emph {g $ g $} $是最小正整数$ k $，因此必须存在一个必须细分的$ k $的边缘，以增加$ k $ timples的$ g $ g $ $ g $ $ g $。在本文中，我们表明$ msd_ {γ_{t2}}（g）\ leq 3 $对于任何图的订单$ g $至少$ 3 $，我们还确定了某些图形类别的半统治多动型数，并用$ t $表征$ t $，并带有$ msd_ {γ_{γ_{γ_{γ_{t2}} $。另一方面，我们知道$γ_{t2}（g）$是一个参数，在统治数字，$γ（g）$和总统治数量，$γ_T（g）$之间挤压，因此对于任何树$ t $ $ \ frac {γ_T（t）} {γ_{t2}（t）} $，并呈现树木家族实现上限的建设性特征。
A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in $S$ is within distance $2$ of another vertex of $S$. The \emph{semitotal domination number}, $γ_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. The \emph{semitotal domination multisubdivision number} of a graph $G$, $msd_{γ_{t2}}(G)$, is the minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to increase the semitotal domination number of $G$. In this paper, we show that $msd_{γ_{t2}}(G)\leq 3$ for any graph $G$ of order at least $3$, we also determine the semitotal domination multisubdivision number for some classes of graphs and characterize trees $T$ with $msd_{γ_{t2}}(T)=3$. On the other hand, we know that $γ_{t2}(G)$ is a parameter that is squeezed between domination number, $γ(G)$ and total domination number, $γ_t(G)$, so for any tree $T$, we investigate the ratios $\frac{γ_{t2}(T)}{γ(T)}$ and $\frac{γ_t(T)}{γ_{t2}(T)}$, and present the constructive characterizations of the families of trees achieving the upper bounds.
