论文标题
有向树连接的极端结果
Extremal results for directed tree connectivity
论文作者
论文摘要
对于Digraph $ d =（v（d），a（d））$，$ s \ subseteq v（d）$带有$ r \ in s $ in s $和$ | s | \ geq 2 $，a $（s，r）$ - tree是$ r $ rug的$ r $ s $ s \ subseteq v（t）。如果$ a（t_1）\ cap a（t_2）= \ emptyset $，则两个$（s，r）$ - 树$ t_1 $和$ t_2 $。如果$ v（t_1）\ cap v（t_2）= s $，则两个Arc-disjoint $（S，R）$ - 树$ T_1 $和$ T_2 $在内部不相交。令$κ_{s，r}（d）$和$λ_{s，r}（d）$分别是内部分离的最大数量和arc-disjoint $（s，r）$ - $ d $中的树。 $ d $ $ d $的广义$ k $ -vertex-strong连接定义为$κ_k（d）= \ min \ {κ__{κ_{k {s，r}（d）\ mid s \ mid s \ subset v（d），| s | s | = k = k，in s \}。 $$λ_k(D)= \min \{λ_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ The generalized $k$-vertex-strong connectivity and generalized $k$-arc-strong connectivity are also called directed tree connectivity which could be seen as a generalization of classical connectivity of digraphs. digraph $ d =（v（d），a（d））$称为最低概括$（k，\ ell）$ - $ - 顶点（分别为，，arc） - 如果$κ__k（d）\ geq \ ell $（分别为$λ_k（$λ_k（d）\ geq \ ell $），则连接$λ_k（d），但$ e e \ e \， $κ_K（d-e）\ leq \ ell-1 $（分别为$λ_k（d-e）\ leq \ ell-1 $）。在本文中，我们研究了最小概括的$（k，\ ell）$ - 顶点（分别为ARC） - 连接的Digraphs。我们计算了这些挖掘物的最小和最大尺寸，并为$ k $和$ \ ell $的某些成对的Digraphs提供了此类挖掘的特征。
For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $κ_{S,r}(D)$ and $λ_{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized $k$-vertex-strong connectivity of $D$ is defined as $$κ_k(D)= \min \{κ_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ Similarly, the generalized $k$-arc-strong connectivity of $D$ is defined as $$λ_k(D)= \min \{λ_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ The generalized $k$-vertex-strong connectivity and generalized $k$-arc-strong connectivity are also called directed tree connectivity which could be seen as a generalization of classical connectivity of digraphs. A digraph $D=(V(D), A(D))$ is called minimally generalized $(k, \ell)$-vertex (respectively, arc)-strongly connected if $κ_k(D)\geq \ell$ (respectively, $λ_k(D)\geq \ell$) but for any arc $e\in A(D)$, $κ_k(D-e)\leq \ell-1$ (respectively, $λ_k(D-e)\leq \ell-1$). In this paper, we study the minimally generalized $(k, \ell)$-vertex (respectively, arc)-strongly connected digraphs. We compute the minimum and maximum sizes of these digraphs, and give characterizations of such digraphs for some pairs of $k$ and $\ell$.
