Locally cotriangular graphs
A graph \(G\) is said to be locally \(X\), where \(X\) is a family of graphs, if for every vertex \(x\) of \(G\), the neighbours of \(x\) span a graph from \(X\). Various theorems concerning locally \(X\) graphs are proved for several classes \(X\). E.g., the main theorem characterizes graphs which are locally \(D\) where \(D\) is the class of all (essentially) finite non-degenerate indecomposable cotriangular graphs. (\(G=(V,E)\) is called indecomposable if it admits no decomposition \(V=A\cup B\), \(A\neq \emptyset \neq B\) such that \(ab\in E\) for all \(a\in A\), \(b\in B\); \(G\) is called cotriangular if for every pair \(x,y\in V\) with \(xy\not\in E\) there exists \(z\in V\) such that \(\{x,y,z\}\) is an independent set and such that every vertex \(v\neq x,y,z\) is adjacent to exactly one or to all of the vertices \(x,y,z\).)
scientific article; zbMATH DE number 3924839