论文标题
普通图中的统治与独立统治
Domination versus independent domination in regular graphs
论文作者
论文摘要
如果$ g $的每个顶点$ s $ s $ s $或与$ s $相邻的顶点，则图$ g $中的$ s $顶点是一个主体集。如果此外，$ s $是独立的集合，则$ s $是独立的主导套装。 $ g $的支配数字$γ（g）$是$ g $中的主要基数的最低基数，而独立的统治数$ i（g）$ g $的$ g $是$ g $中独立支配的最低基数。我们证明，对于所有整数$ k \ geq 3 $，它认为，如果$ g $是连接的$ k $ - 台式图，则$ \ frac {i（g）} {γ（g）} \ leq \ leq \ frac {k} {k} {2} {2} {2} $，仅具有等于$ g = k = k_ k _ k _ k_ k} $。结果以前仅以$ k \ leq 6 $而闻名。这肯定回答了Babikir和Henning的最新问题。
A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $G$ is in $S$ or is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The domination number $γ(G)$ of $G$ is the minimum cardinality of a dominating set in $G$, while the independent domination number $i(G)$ of $G$ is the minimum cardinality of an independent dominating set in $G$. We prove that for all integers $k \geq 3$ it holds that if $G$ is a connected $k$-regular graph, then $\frac{i(G)}{γ(G)} \leq \frac{k}{2}$, with equality if and only if $G = K_{k,k}$. The result was previously known only for $k\leq 6$. This affirmatively answers a recent question of Babikir and Henning.
