论文标题
关于树木中共同独立统治的注释
A note on total co-independent domination in trees
论文作者
论文摘要
如果每个顶点$ g $的每个顶点都与至少一个$ d $的顶点相邻，则图$ g $的顶点$ d $是总的统治集。 $ g $的总统治数是$ g $的任何总统治集的最低基数，并用$γ_T（g）$表示。如果$ v（g）\ setminus d $是独立集，并且至少具有一个顶点，则总统治集合$ d $称为共同独立的统治集。 $γ_{t，coi}（g）$表示，任何共同独立的主导集的最小基数。在本文中，我们表明，对于任何树$ t $ t $ n $和直径至少三个，$ n-β（t）\leqγ_{t，coi}（t）（t）\ leq n- | l（t）| $β（t）$β（t）$是任何独立的套装的最大基础，$ l（t）$是$ l（t）$是$ t $ t $ t $ t $ t $。我们还表征了达到上述极端界限的树木家属，并表明$γ_{t，coi}（t）$的价值与某些树类的界限之间的差异可能是任意的。
A set $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total domination number of $G$ is the minimum cardinality of any total dominating set of $G$ and is denoted by $γ_t(G)$. The total dominating set $D$ is called a total co-independent dominating set if $V(G)\setminus D$ is an independent set and has at least one vertex. The minimum cardinality of any total co-independent dominating set is denoted by $γ_{t,coi}(G)$. In this paper, we show that, for any tree $T$ of order $n$ and diameter at least three, $n-β(T)\leq γ_{t,coi}(T)\leq n-|L(T)|$ where $β(T)$ is the maximum cardinality of any independent set and $L(T)$ is the set of leaves of $T$. We also characterize the families of trees attaining the extremal bounds above and show that the differences between the value of $γ_{t,coi}(T)$ and these bounds can be arbitrarily large for some classes of trees.
