Room : P516 

Time/Day : 14:00 of 17/06/2026

Title: Neighbourhood convexity in graphs

Abstract:

While initially developed in Analysis and Geometry, the notion of convexity has been a fruitful one in many areas of mathematics ([4]). In particular, finite convexities have a great deal of inuence in Graph Theory because allow to characterize many classes of graphs such as chordal graphs and ptolemaic graphs (see [3] for a survey of diiferent types of convexities in graphs).

I am going to introduce a new convexity in graphs, called neighbourhood convexity which has never been addressed in the literature. In fact it will be obtained from a Moore closure operator that was fruitfully conceived by Bubboloni and Pinzauti in order to deal with power graphs of finite groups (see [1] and [2]).

The neighbourhood convexity is based on the concept of common neighbourhood of a subset of vertices instead of, as usual, in paths among vertices of the graph.

We explore properties of graphs G = (V;E) for which the neighbourhood convexity is a convex geometry observing, among other things, that neighbourhood convexity is an example of convex geometry that is almost never hereditary. From the fact that no pair of true twins can exist in a neighbourhood convex geometry G, we

associate with G a poset P(G) and link convex sets of G with upsets of P(G), as well as extreme vertices of convex sets of G with minimal elements in their induced poset in P(G).

This research is a joint work with José Caceres.

References:

[1] D. Bubboloni, N. Pinzauti, Critical classes of power graphs and reconstruction of directed

power graphs, Journal of Group Theory 28 (2025), 713{739.

[2] D. Bubboloni, N. Pinzauti, Critical groups and partitions of finite groups, Mediterranean

Journal of Mathematics 22, 131 (2025). doi.org/10.1007/s00009-025-02865-8

[3] M.C. Dourado, M. Guitierrez, F. Protti, R. Sampaio, S. Tondato, Characterizations of graph

classes via convex geometries: A survey, Discrete Applied Mathematics 360 (2025) pp. 246{257.

[4] M. Van de Vel, Theory of Convex Structures, North-Holland, Amsterdam, MA, 1993.