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Strong Subgraph Connectivity of Digraphs
Graphs and Combinatorics ( IF 0.6 ) Pub Date : 2021-03-18 , DOI: 10.1007/s00373-021-02294-w
Yuefang Sun , Gregory Gutin

Let \(D=(V,A)\) be a digraph of order n, S a subset of V of size k and \(2\le k\le n\). A strong subgraph H of D is called an S-strong subgraph if \(S\subseteq V(H)\). A pair of S-strong subgraphs \(D_1\) and \(D_2\) are said to be arc-disjoint if \(A(D_1)\cap A(D_2)=\emptyset\). A pair of arc-disjoint S-strong subgraphs \(D_1\) and \(D_2\) are said to be internally disjoint if \(V(D_1)\cap V(D_2)=S\). Let \(\kappa _S(D)\) (resp. \(\lambda _S(D)\)) be the maximum number of internally disjoint (resp. arc-disjoint) S-strong subgraphs in D. The strong subgraph k -connectivity is defined as

$$\begin{aligned} \kappa _k(D)=\min \{\kappa _S(D)\mid S\subseteq V, |S|=k\}. \end{aligned}$$

As a natural counterpart of the strong subgraph k-connectivity, we introduce the concept of strong subgraph k -arc-connectivity which is defined as

$$\begin{aligned} \lambda _k(D)=\min \{\lambda _S(D)\mid S\subseteq V(D), |S|=k\}. \end{aligned}$$

A digraph \(D=(V, A)\) is called minimally strong subgraph \((k,\ell )\)-(arc-)connected if \(\kappa _k(D)\ge \ell\) (resp. \(\lambda _k(D)\ge \ell\)) but for any arc \(e\in A\), \(\kappa _k(D-e)\le \ell -1\) (resp. \(\lambda _k(D-e)\le \ell -1\)). In this paper, we first give complexity results for \(\lambda _k(D)\), then obtain some sharp bounds for the parameters \(\kappa _k(D)\) and \(\lambda _k(D)\). Finally, minimally strong subgraph \((k,\ell )\)-connected digraphs and minimally strong subgraph \((k,\ell )\)-arc-connected digraphs are studied.



中文翻译:

有向图的强大子图连通性

\(D =(V,A)\)n阶的有向图,S是大小为kV的子集和\(2 \ le k \ le n \)。甲强子图ħd被称为小号-子图,如果\(S \ subseteq V(1H)\) 。如果\(A(D_1)\ cap A(D_2)= \ emptyset \),则一对S -strong子图\(D_1 \)\(D_2 \)被称为弧不相交的。一对弧形不相交的S-强子图\(D_1 \)\(D_2 \)如果\(V(D_1)\ cap V(D_2)= S \)被称为内部不相交。让\(\卡帕_S(d)\) (相应地,\(\拉姆达_S(d)\) )在内部不相交的最大数目(分别为圆弧不相交)š在-strong子图d。所述强子图ķ -connectivity被定义为

$$ \ begin {aligned} \ kappa _k(D)= \ min \ {\ kappa _S(D)\ mid S \ subseteq V,| S | = k \}。\ end {aligned} $$

作为强子图k-连通性的自然对应物,我们介绍了强子图 k -arc-连通性的概念,其定义为

$$ \ begin {aligned} \ lambda _k(D)= \ min \ {\ lambda _S(D)\ mid S \ subseteq V(D),| S | = k \}。\ end {aligned} $$

有向图\(D =(V,A)\)被称为最小强子图 \((k,\ ell)\)-(arc-)如果\(\ kappa _k(D)\ ge \ ell \)(分别为\(\ lambda _k(D)\ ge \ ell \)),但对于任何弧线\(e \ in A \)\(\ kappa _k(De)\ le \ ell -1 \)(resp。\ (\ lambda _k(De)\ le \ ell -1 \))。在本文中,我们首先给出\(\ lambda _k(D)\)的复杂度结果,然后为参数\(\ kappa _k(D)\)\(\ lambda _k(D)\)获得一些尖锐边界。最后,最小强子图\((k,\ ell)\)相连的有向图和最小强子图\((k,\ ell)\)-连接弧的有向图。

更新日期:2021-03-19
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