A new lower bound for the independent domination number of a tree

Abel Cabrera Martínez
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Abstract

A set $D$ of vertices in a graph $G$ is an independent dominating set of $G$ if $D$ is an independent set and every vertex not in $D$ is adjacent to a vertex in $D$. The independent domination number of $G$, denoted by $i(G)$, is the minimum cardinality among all independent dominating sets of $G$. In this paper we show that if $T$ is a nontrivial tree, then $i(T)\geq \frac{n(T)+\gamma(T)-l(T)+2}{4}$, where $n(T)$, $\gamma(T)$ and $l(T)$ represent the order, the domination number and the number of leaves of $T$, respectively. In addition, we characterize the trees achieving this new lower bound.
树的独立支配数的新下界
如果$D$是一个独立的集合,并且不在$D$中的每个顶点都与$D$中的一个顶点相邻,那么图$G$中的顶点集$D$就是$G$的独立支配集。$G$的独立支配数用$i(G)$表示,它是$G$的所有独立支配集的最小基数。本文证明了如果$T$是一棵非平凡树,那么$i(T)\geq \frac{n(T)+\gamma(T)-l(T)+2}{4}$,其中$n(T)$、$\gamma(T)$和$l(T)$分别表示$T$的阶数、支配数和叶数。此外,我们描述了实现这个新的下界的树。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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