Characterization and recognition of generalized clique-Helly graphs

Let p?1 and q?0 be integers. A family of sets F is (p,q)-intersecting when every subfamily F^′⊆F formed by p or less members has total intersection of cardinality at least q. A family of sets F is (p,q)-Helly when every (p,q)-intersecting subfamily F^′⊆F has total intersection of cardinality at least q. A graph G is a (p,q)-clique-Helly graph when its family of (maximal) cliques is (p,q)-Helly. According to this terminology, the usual Helly property and the clique-Helly graphs correspond to the case p=2,q=1. In this work we present a characterization for (p,q)-clique-Helly graphs. For fixed p,q, this characterization leads to a polynomial-time recognition algorithm. When p or q is not fixed, it is shown that the recognition of (p,q)-clique-Helly graphs is NP-hard.     

(Co)oxidation/cyclization processes upon irradiation of triphenylamine

Irradiation of triphenylamine (Ph₃N) in nitrogen-flushed solution leads to 9-phenylcarbazole and two tetrahydroderivatives (1,2,3,4- and 1,2,7,8-) via disproportionation of the corresponding 4a,4b-dihydrocarbazole. In oxygen-equilibrated solution oxidative cyclization occurs through the intermediacy of a triplet peroxy diradical, which either abstracts a hydrogen atom intramolecularly or (mainly) cleaves back to the reagents. The role of the key intermediates is supported by DFT calculations and by trapping by triarylphosphines (that are thus efficiently oxidized, while preventing the cyclization of Ph₃N). The hydroperoxide, on the other hand, causes inefficient co-oxidation of sulfides.     

Diradicals Photogeneration from Chloroaryl-Substituted Carboxylic Acids

With the aim of generating new, thermally inaccessible diradicals, potentially able to induce a double-strand DNA cleavage, the photochemistry of a set of chloroaryl-substituted carboxylic acids in polar media was investigated. The photoheterolytic cleavage of the Ar−Cl bond occurred in each case to form the corresponding triplet phenyl cations. Under basic conditions, the photorelease of the chloride anion was accompanied by an intramolecular electron-transfer from the carboxylate group to the aromatic radical cationic site to give a diradical species. This latter intermediate could then undergo CO₂ loss in a structure-dependent fashion, according to the stability of the resulting diradical, or abstract a hydrogen atom from the medium. In aqueous environment at physiological pH (pH=7.3), both a phenyl cation and a diradical chemistry was observed. The mechanistic scenario and the role of the various intermediates (aryl cations and diradicals) involved in the process was supported by computational analysis.     

Algorithmic aspects of Steiner convexity and enumeration of Steiner trees

For a set \(W\) of vertices of a connected graph \(G=(V(G),E(G))\) , a Steiner W-tree is a connected subgraph \(T\) of \(G\) such that \(W\subseteq V(T)\) and \(|E(T)|\) is minimum. Vertices in \(W\) are called terminals. In this work, we design an algorithm for the enumeration of all Steiner \(W\) -trees for a constant number of terminals, which is the usual scenario in many applications. We discuss algorithmic issues involving space requirements to compactly represent the optimal solutions and the time delay to generate them. After generating the first Steiner \(W\) -tree in polynomial time, our algorithm enumerates the remaining trees with \(O(n)\) delay (where \(n=|V(G)|\) ). An algorithm to enumerate all Steiner trees was already known (Khachiyan et al., SIAM J Discret Math 19:966–984, 2005), but this is the first one achieving polynomial delay. A by-product of our algorithm is a representation of all (possibly exponentially many) optimal solutions using polynomially bounded space. We also deal with the following problem: given \(W\) and a vertex \(x\in V(G)\setminus W\) , is \(x\) in a Steiner \(W'\) -tree for some \(\emptyset \ne W' \subseteq W\) ? This problem is investigated from the complexity point of view. We prove that it is NP-hard when \(W\) has arbitrary size. In addition, we prove that deciding whether \(x\) is in some Steiner \(W\) -tree is NP-hard as well. We discuss how these problems can be used to define a notion of Steiner convexity in graphs.     

MRI-guided neutron capture therapy by use of a dual gadolinium/boron agent targeted at tumour cells through upregulated low-density lipoprotein transporters

The upregulation of low-density lipoprotein (LDL) transporters in tumour cells has been exploited to deliver a sufficient amount of gadolinium/boron/ligand (Gd/B/L) probes for neutron capture therapy, a binary chemio-radiotherapy for cancer treatment. The Gd/B/L probe consists of a carborane unit (ten B atoms) bearing an aliphatic chain on one side (to bind LDL particles), and a Gd(III)/1,4,7,10-tetraazacyclododecane monoamide complex on the other (for detection by magnetic resonance imaging (MRI)). Up to 190 Gd/B/L probes were loaded per LDL particle. The uptake from tumour cells was initially assessed on cell cultures of human hepatoma (HepG2), murine melanoma (B16), and human glioblastoma (U87). The MRI assessment of the amount of Gd/B/L taken up by tumour cells was validated by inductively coupled plasma-mass-spectrometric measurements of the Gd and B content. Measurements were undertaken in vivo on mice bearing tumours in which B16 tumour cells were inoculated at the base of the neck. From the acquisition of magnetic resonance images, it was established that after 4-6 hours from the administration of the Gd/B/L-LDL particles (0.1 and 1 mmol kg(-1) of Gd and (10)B, respectively) the amount of boron taken up in the tumour region is above the threshold required for successful NCT treatment. After neutron irradiation, tumour growth was followed for 20 days by MRI. The group of treated mice showed markedly lower tumour growth with respect to the control group.     

On the strong p-Helly property

The notion of strong p-Helly hypergraphs was introduced by Golumbic and Jamison in 1985 [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, J. Combin. Theory Ser. B 38 (1985) 8-22]. Independently, other authors [A. Bretto, S. Ubéda, J. ?erovnik, A polynomial algorithm for the strong Helly property. Inform. Process. Lett. 81 (2002) 55-57, E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220, W.D. Wallis, Guo-Hui Zhang, On maximal clique irreducible graphs. J. Combin. Math. Combin. Comput. 8 (1990) 187-193.] have also considered the strong Helly property in other contexts. In this paper, we characterize strong p-Helly hypergraphs. This characterization leads to an algorithm for recognizing such hypergraphs, which terminates within polynomial time whenever p is fixed. In contrast, we show that the recognition problem is co-NP-complete, for arbitrary p. Further, we apply the concept of strong p-Helly hypergraphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs. For p=2, this class is equivalent to that of hereditary clique-Helly graphs [E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220]. We describe a characterization for this class and obtain an algorithm for recognizing such graphs. Again, the algorithm has polynomial-time complexity for p fixed, and we show the corresponding recognition problem to be NP-hard, for arbitrary p.     

On the complete set packing and set partitioning polytopes: Properties and rank 1 facets

This paper studies two polytopes: the complete set packing and set partitioning polytopes, which are both associated with a binary $n$-row matrix having all possible columns. Cuts of rank 1 for the latter polytope play a central role in recent exact algorithms for many combinatorial problems, such as vehicle routing. We show the precise relation between the two polytopes studied, characterize the multipliers that induce rank 1 clique facets and give several families of multipliers that yield other facets.     

Phosphate esters as "tunable" reagents in organic synthesis

The essential role phosphates have in biochemistry has no parallel in man-made chemistry, where the poor reactivity of these esters towards nucleophilic substitution makes more reactive substrates such as halides and sulfonates preferred. Nevertheless, phosphates are acquiring an increasing role in organic synthesis as long as new activation modes become available. These include metal catalysis and photochemistry (the latter effective with benzyl and aryl derivatives) that may turn the present limitation into a promising opportunity by devising convenient tunable reactions.