On constructible graphs, locally Helly graphs, and convexity

A (finite or infinite) graph G is constructible if there exists a well-ordering of its vertices such that for every vertex x which is not the smallest element, there is a vertex y < x which is adjacent to x and to every neighbor z of x with z < x . Particular constructible graphs are Helly graphs and connected bridged graphs. In this paper we study a new class of constructible graphs, the class of locally Helly graphs. A graph G is locally Helly if, for every pair (x,y ) of vertices of G whose distance is d 2, there exists a vertex whose distance to x is d 1 and which is adjacent to y and to all neighbors of y whose distance to x is at most d. Helly graphs are locally Helly, and the converse holds for finite graphs. Among different properties we prove that a locally Helly graph is strongly dismantable, hence cop-win, if and only if it contains no isometric rays. We show that a locally Helly graph G is finitely Helly, that is, every finite family of pairwise