Towards obtaining a 3-Decomposition from a perfect Matching

A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph, and a matching. It has been settled for special classes of graphs, one of the first results being for Hamiltonian graphs. In the past two years several new results have been obtained, adding the classes of plane, claw-free, and 3-connected tree-width 3 graphs to the list. In this paper, we regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching $M$ from a cubic graph $G$ leaves a disjoint union of cycles. Contracting these cycles yields a new graph $G_M$. The graph $G$ is star-like if $G_M$ is a star for some perfect matching $M$, making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well.Comment: 21 pages, 7 figure

Semi-Preemptive Routing on Trees

We study a variant of the pickup-and-delivery problem (PDP) in which the objects that have to be transported can be reloaded at most d times, for a given integer d. This problem is known to be polynomially solvable on paths or cycles and NP-complete on trees. We present a (4/3+epsilon)-approximation algorithm if the underlying graph is a tree. By using a result of Charikar et al. (1998), this can be extended to a O(log n log log n)-approximation for general graphs

Semi-Preemptive Routing on Trees

We study a variant of the pickup-and-delivery problem (PDP) in which the objects that have to be transported can be reloaded at most d times, for a given integer d. This problem is known to be polynomially solvable on paths or cycles and NP-complete on trees. We present a (4/3+epsilon)-approximation algorithm if the underlying graph is a tree. By using a result of Charikar et al. (1998), this can be extended to a O(log n log log n)-approximation for general graphs