Convexity in Helly Graphs: Selection and Almost Fixed Point Properties for Multifunctions

In this paper, a notion of convexity structure for graphs, called neighbourhood convexity, is introduced. It is shown that neighbourhood convexity is exactly the graph convexity for Helly graphs to be 2-Helly. We then consider various neighbourhood-convexity almost fixed point properties for Helly graphs. In particular, we show that for any positive number p, every Helly graph has the neighbourhood convexity -almost fixed point property for p-weak multifunctions; and any self-mapping neighbourhood convexity strong multifunction has a selection.