Locally cotriangular graphs

Summary

A graph G is said to be locally X, where X is any collection of graphs if, for each vertex x in G, the subgraph (x) of all vertices of G adjacent to x is isomorphic to some member of X. In the case that X consists of a single graph X, we say G is locally X instead of locally {X}.

A graph G is said to possess the cotriangle property if for every pair (x, y) of non-adjacent vertices in G there exists a third vertex z forming a subgraph T = {x, y, z} isomorphic to K₃ (that is, a “cotriangle”) having the property that any vertex u of G not lying in the cotriangle T is adjacent to exactly one or all of the vertices of T.

We write G = A + B, if A and B are disjoint subgraphs of G the union of whose vertices comprise the vertices of G, and if every vertex in A is adjacent to every vertex in B. Such a graph G possesses the cotriangle property if and only if both A and B do. We say G is indecomposable if it admits no such decomposition G = A + B into non-empty subgraphs A and B.