The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

Electrografting of calix[4]arenediazonium salts to form versatile robust platforms for spatially controlled surface functionalization

An essential issue in the development of materials presenting an accurately functionalized surface is to achieve control of layer structuring. Whereas the very popular method based on the spontaneous adsorption of alkanethiols on metal faces stability problems, the reductive electrografting of aryldiazonium salts yielding stable interface, struggles with the control of the formation and organization of monolayers. Here we report a general strategy for patterning surfaces using aryldiazonium surface chemistry. Calix[4]tetra-diazonium cations generated in situ from the corresponding tetra-anilines were electrografted on gold and carbon substrates. The well-preorganized macrocyclic structure of the calix[4]arene molecules allows the formation of densely packed monolayers. Through adequate decoration of the small rim of the calixarenes, functional molecules can then be introduced on the immobilized calixarene subunits, paving the way for an accurate spatial control of the chemical composition of a surface at molecular level.SCOPUS: ar.jinfo:eu-repo/semantics/publishe