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arXiv 1502.06245math.CO
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图的控制数的变化：边多重剖分与边去除

Changing of the domination number of a graph: edge multisubdivision and edge removal

Vladimir Samodivkin

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11 pages
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摘要：

对于一个图形性质$\mathcal{P}$和一个图$G$，如果由$S$导出的子图具有该性质$\mathcal{P}$，则$G$的顶点子集$S$是一个$\mathcal{P}$ -集.关于性质$\mathcal{P}$的控制数，记为$γ_{\mathcal{P}} (G)$，是控制$\mathcal{P}$ -集的最小基数。定义了$\mathcal{P}$的控制多重细分数，记为$msd_{\mathcal{P}}(G)$，表示为一个最小正整数$k$，使得存在一条边，该边必须细分$k$次才能改变$γ_\mathcal{P} (G)$ .本文（a）给出了$G$的边剖分一次后$γ_{\mathcal{P}}(G)$变化的充要条件，（b）证明了：如果$e$是图$G$的边，则$γ_\mathcal{P} (G_{e,1}) < γ_\mathcal{P} (G)$当且仅当$γ_\mathcal{P} (G-e) < γ_\mathcal{P} (G)$（$G_{e,t}$表示通过用$t$顶点细分$e$而从$G$获得的图），（c）证明了对图的每条边，$G$满足$γ_{\mathcal{P}}(G-e) \leq γ_{\mathcal{P}}(G_{e,3}) \leq γ_{\mathcal{P}}(G-e) + 1$;（d）证明了图的$msd_{\mathcal{P}}(G) \leq 3$，其中$\mathcal{P}$是遗传的，且在与$K_1$的并下是闭的.

英文摘要：

For a graphical property $\mathcal{P}$ and a graph $G$, a subset $S$ of vertices of $G$ is a $\mathcal{P}$-set if the subgraph induced by $S$ has the property $\mathcal{P}$. The domination number with respect to the property $\mathcal{P}$, denoted by $γ_{\mathcal{P}} (G)$, is the minimum cardinality of a dominating $\mathcal{P}$-set. We define the domination multisubdivision number with respect to $\mathcal{P}$,denoted by $msd_{\mathcal{P}}(G)$, as a minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to change $γ_\mathcal{P} (G)$. In this paper (a) we present necessary and sufficient conditions for a change of $γ_{\mathcal{P}}(G)$ after subdividing an edge of $G$ once, (b) we prove that if $e$ is an edge of a graph $G$ then $γ_\mathcal{P} (G_{e,1}) < γ_\mathcal{P} (G)$ if and only if $γ_\mathcal{P} (G-e) < γ_\mathcal{P} (G)$ ($G_{e,t}$ denote the graph obtained from $G$ by subdivision of $e$ with $t$ vertices), (c) we also prove that for every edge of a graph $G$ is fulfilled $γ_{\mathcal{P}}(G-e) \leq γ_{\mathcal{P}}(G_{e,3}) \leq γ_{\mathcal{P}}(G-e) + 1$, and (d) we show that $msd_{\mathcal{P}}(G) \leq 3$, where $\mathcal{P}$ is hereditary and closed under union with $K_1$.

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