Line-distortion, Bandwidth and Path-length of a graph

We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson-Seymour's path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show: - if a graph $G$ can be embedded into the line with distortion $k$, then $G$ admits a Robertson-Seymour's path-decomposition with bags of diameter at most $k$ in $G$; - for every class of graphs with path-length bounded by a constant, there exist an efficient constant-factor approximation algorithm for the minimum line-distortion problem and an efficient constant-factor approximation algorithm for the minimum bandwidth problem; - there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph; - AT-free graphs and some intersection families of graphs have path-length at most 2; - for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum line-distortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem

Complexity Results for the Spanning Tree Congestion Problem

We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the complexity of this problem. First, we show that for every fixed k and d the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k ≥ 10. For very small values of k however, the problem becomes polynomially solvable. We also show that it is NP-hard to approximate the spanning tree congestion within a factor better than 11/10. On planar graphs, we prove the problem is NP-hard in general, but solvable in linear time for fixed k

Sviluppo pre-clinico e clinico di inibitori della cellula staminale leucemica nelle leucemie acute

In Leukemias, recent developments have demonstrated that the Hedgehog pathway plays a key-role in the peculiar ability of self renewal of leukemia stem cells. The aim of this research activity was to investigate, through a first in man, Phase I, open label, clinical trial, the role and the impact, mainly in terms of safety profile, adverse events and pharmacokinetics, of a Sonic Hedgehog inhibitor compound on a population of heavely pretreated patients affected by AML, CML, MF, or MDS, resistant or refractory to standard chemotherapy. Thirty-five patients have been enrolled. The drug was administered orally, in 28 days cycles, without rest periods. The compound showed a good safety profile. The half life was of 17-35 hours, justifying the daily administration. Significant signs of activity, in terms of reduction of bone marrow blast cell amount were seen in most of the patients enrolled. Interestingly, correlative biological studies demonstrated that, comparing the gene expression profyiling signature of separated CD34+ cells before and after one cycle of treatment, the most variably expressed genes were involved in the Hh pathway. Moreover, we observed that many genes involved in MDR (multidrug resistance)were significantly down regulated after treatment. These data might lead to future clinical trials based on combinatory approaches, including, for instance, Hh inhibitors and conventional chemotherapy

On the power of bfs to determine a graph’s diameter

Abstract. Recently considerable effort has been spent on showing that Lexicographic Breadth First Search (LBFS) can be used to determine a tight bound on the diameter of graphs from various restricted classes. In this paper, we show that in some cases, the full power of LBFS is not required and that other variations of Breadth First Search (BFS) suffice. The restricted graph classes that are amenable to this approach all have a small constant upper bound on the maximum sized cycle that may appear as an induced subgraph. We show that on graphs that have no induced cycle of size greater than k, BFS finds an estimate of the diameter that is no worse than diam(G) −⌊k/2⌋−2.

Navigating in a graph by aid of its spanning tree

Let G =(V,E) be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, z = x), unless z = y, gotoaneighborofz in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighborhood in G and can use the distances in T to navigate in G. Thus, additionally to standard local information (the neighborhood NG(v)), the only global information that is available to each vertex v is the topology of the spanning tree T (in fact, v can know only a very small piece of information about T and still be able to infer from it the necessary tree-distances). For each source vertex x and target vertex y, this way, a path, called a greedy routing path, is produced. Denote by gG,T (x, y) the length of a longest greedy routing path that can be produced for x and y using this strategy and T. We say that a spanning tree T of a graph G is an additive r-carcass for G if gG,T (x, y) ≤ dG(x, y)+r for each ordered pair x, y ∈ V. In this paper, we investigate the problem, given a graph family F, whether a small integer r exists such that any graph G ∈Fadmits an additive r-carcass. We show that rectilinear p × q grids, hypercubes, distance-hereditary graphs, dually chordal graphs (and, therefore, strongly chordal graphs and interval graphs), all admit additive 0-carcasses. Furthermore, every chordal graph G admits an additive (ω(G) + 1)-carcass (where ω(G) is the size of a maximum clique of G), each 3-sun-free chordal graph admits an additive 2-carcass, each chordal bipartite graph admits an additive 4-carcass. In particular, any k-tree admits an additive (k+2)-carcass. All those carcasses are easy to construct