Generalizations of the Tree Packing Conjecture

The Gy\'arf\'as tree packing conjecture asserts that any set of trees with $2,3, ..., k$ vertices has an (edge-disjoint) packing into the complete graph on $k$ vertices. Gy\'arf\'as and Lehel proved that the conjecture holds in some special cases. We address the problem of packing trees into $k$-chromatic graphs. In particular, we prove that if all but three of the trees are stars then they have a packing into any $k$-chromatic graph. We also consider several other generalizations of the conjecture

The Planar Tree Packing Theorem

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garc\'ia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.Comment: Full version of our SoCG 2016 pape

The Complexity of Packing Edge-Disjoint Paths

We introduce and study the complexity of Path Packing. Given a graph G and a list of paths, the task is to embed the paths edge-disjoint in G. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard. Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case Exact Path Packing where the paths have to cover every edge in G exactly once is already NP-hard for two paths on 4-regular graphs

Cut Tree Construction from Massive Graphs

The construction of cut trees (also known as Gomory-Hu trees) for a given graph enables the minimum-cut size of the original graph to be obtained for any pair of vertices. Cut trees are a powerful back-end for graph management and mining, as they support various procedures related to the minimum cut, maximum flow, and connectivity. However, the crucial drawback with cut trees is the computational cost of their construction. In theory, a cut tree is built by applying a maximum flow algorithm for $n$ times, where $n$ is the number of vertices. Therefore, naive implementations of this approach result in cubic time complexity, which is obviously too slow for today's large-scale graphs. To address this issue, in the present study, we propose a new cut-tree construction algorithm tailored to real-world networks. Using a series of experiments, we demonstrate that the proposed algorithm is several orders of magnitude faster than previous algorithms and it can construct cut trees for billion-scale graphs.Comment: Short version will appear at ICDM'1

The generalized 3-edge-connectivity of lexicographic product graphs

The generalized $k$-edge-connectivity $\lambda_k(G)$ of a graph $G$ is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs $G$ and $H$, denoted by $G\circ H$, is an important graph product. In this paper, we mainly study the generalized 3-edge-connectivity of $G \circ H$, and get upper and lower bounds of $\lambda_3(G \circ H)$. Moreover, all bounds are sharp.Comment: 14 page