Independent branchings in acyclic diagraphs

AbstractLet D be a finite directed acyclic multigraph and t be a vertex of D such that for each other vertex x of D, there are n pairwise openly disjoint paths in D from x to t. It is proved that there exist n spanning trees B1,…,Bn in D directed toward t such that for each vertex x ≠ t of D, the n paths from x to t in B1,…,Bn are pairwise openly disjoint

Dominators in Directed Graphs: A Survey of Recent Results, Applications, and Open Problems

The computation of dominators is a central tool in program optimization and code generation, and it has applications in other diverse areas includingconstraint programming, circuit testing, and biology. In this paper we survey recent results, applications, and open problems related to the notion of dominators in directed graphs,including dominator verification and certification, computing independent spanning trees, and connectivity and path-determination problems in directed graphs

Completely Independent Spanning Trees in Line Graphs

Completely independent spanning trees in a graph $G$ are spanning trees of $G$ such that for any two distinct vertices of $G$, the paths between them in the spanning trees are pairwise edge-disjoint and internally vertex-disjoint. In this paper, we present a tight lower bound on the maximum number of completely independent spanning trees in $L(G)$, where $L(G)$ denotes the line graph of a graph $G$. Based on a new characterization of a graph with $k$ completely independent spanning trees, we also show that for any complete graph $K_n$ of order $n \geq 4$, there are $\lfloor \frac{n+1}{2} \rfloor$ completely independent spanning trees in $L(K_n)$ where the number $\lfloor \frac{n+1}{2} \rfloor$ is optimal, such that $\lfloor \frac{n+1}{2} \rfloor$ completely independent spanning trees still exist in the graph obtained from $L(K_n)$ by deleting any vertex (respectively, any induced path of order at most $\frac{n}{2}$) for $n = 4$ or odd $n \geq 5$ (respectively, even $n \geq 6$). Concerning the connectivity and the number of completely independent spanning trees, we moreover show the following, where $\delta(G)$ denotes the minimum degree of $G$. $\$ $\bullet$ Every $2k$-connected line graph $L(G)$ has $k$ completely independent spanning trees if $G$ is not super edge-connected or $\delta(G) \geq 2k$. $\$ $\bullet$ Every $(4k-2)$-connected line graph $L(G)$ has $k$ completely independent spanning trees if $G$ is regular. $\$ $\bullet$ Every $(k^2+2k-1)$-connected line graph $L(G)$ with $\delta(G) \geq k+1$ has $k$ completely independent spanning trees.Comment: 20 pages with 5 figure

Finding multiple maximally redundant trees in linear time

Redundant trees are directed spanning trees, which provide disjoint paths towards their roots. Therefore, this concept is widely applied in the literature both for providing protection and load sharing. The fastest algorithm can find multiple redundant trees, a pair of them rooted at each vertex, in linear time. Unfortunately, edge- or vertex-redundant trees can only be found in 2-edge- or 2-vertex-connected graphs respectively. Therefore, the concept of maximally redundant trees was introduced, which can overcome this problem, and provides maximally disjoint paths towards the common root. In this paper, we propose the first linear time algorithm, which can compute a pair of maximally redundant trees rooted at not only one, but at each vertex

Finding four independent trees

Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2-connected graph. Cheriyan and Maheshwari gave an O(vertical bar V vertical bar(2)) algorithm for finding three independent spanning trees in a 3-connected graph. In this paper we present an O(vertical bar V vertical bar(3)) algorithm for finding four independent spanning trees in a 4-connected graph. We make use of chain decompositions of 4-connected graphs.3551023105