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arXiv 1507.02875math.CO
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弱无$S(K_{1,3})$森林的3/5猜想

The 3/5-conjecture for weakly $S(K_{1,3})$-free forests

Simon Schmidt

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摘要：

关于控制对策的$3/5$ -猜想指出，无孤立点图$G$在$n$顶点上的对策控制数如下有界：$γ_g(G)\leq \frac{3n}5 $和$γ_g'(G)\leq \frac{3n+2}5 $。最近的进展已经完成的主题和猜想现在证明的最小度至少$2$的图。Bujtás介绍的一个强大的工具是\D的贪婪策略。特别是，使用这种策略，她已经证明了在距离$4$没有叶子的孤立森林的猜想。在本文中，我们改进了这一策略，将结果推广到更大的类弱$S(K_{1,3})$ -自由森林，其中弱$S(K_{1,3})$ -自由森林$F$是一个孤立的自由森林没有诱导$S(K_{1,3})$，其叶也是$F$的叶。

英文摘要：

The $3/5$-conjecture for the domination game states that the game domination numbers of an isolate-free graph $G$ on $n$ vertices are bounded as follows: $γ_g(G)\leq \frac{3n}5 $ and $γ_g'(G)\leq \frac{3n+2}5 $. Recent progress have been done on the subject and the conjecture is now proved for graphs with minimum degree at least $2$. One powerful tool, introduced by Bujtás is the so-called greedy strategy for \D. In particular, using this strategy, she has proved the conjecture for isolate-free forests without leafs at distance $4$. In this paper, we improve this strategy to extend the result to the larger class of weakly $S(K_{1,3})$-free forests, where a weakly $S(K_{1,3})$-free forest $F$ is an isolate-free forest without induced $S(K_{1,3})$, whose leafs are leafs of $F$ as well.

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