Oriented trees and paths in digraphs

Which conditions ensure that a digraph contains all oriented paths of some given length, or even a all oriented trees of some given size, as a subgraph? One possible condition could be that the host digraph is a tournament of a certain order. In arbitrary digraphs and oriented graphs, conditions on the chromatic number, on the edge density, on the minimum outdegree and on the minimum semidegree have been proposed. In this survey, we review the known results, and highlight some open questions in the area

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

Mader's conjecture is true for path-star trees

W. Mader [J graph Theory 65 (2010),61-69] conjectured that for any tree $T$ of order $t$, every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor+t-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ is $k$-connected. The conjecture is true when $T$ is a path; $k=1$; partially $k=2$. In this paper, we show that Mader's conjecture is true when $T$ is a path-star.Comment: 8 page

Connectivity keeping spiders in k-connected graphs

W. Mader [J. Graph Theory 65 (2010), 61--69] conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with $\delta(G)\geq\lfloor\frac{3k}{2}\rfloor+m-1$ contains a tree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In this paper, we confirm Mader's conjecture for all the spider-trees.Comment: 9 page

Introduction to Graph Theory

Higher Education Mathematic