Characterizing graph classes by intersections of neighborhoods

The interplay between maxcliques (maximal cliques) and intersections of closed neighborhoods leads to new types of characterizations of several standard graph classes. For instance, being hereditary clique-Helly is equivalent to every nontrivial maxclique $Q$ containing the intersection of closed neighborhoods of two vertices of $Q$, and also to, in all induced subgraphs, every nontrivial maxclique containing a simplicial edge (an edge in a unique maxclique). Similarly, being trivially perfect is equivalent to every maxclique $Q$ containing the closed neighborhood of a vertex of $Q$, and also to, in all induced subgraphs, every maxclique containing a simplicial vertex. Maxcliques can be generalized to maximal cographs, yielding a new characterization of ptolemaic graphs

A Superclass of Edge-Path-Tree graphs with few cliques

Edge-Path-Tree graphs are intersection graphs of Edge-Path-Tree matrices that is matrices whose columns are incidence vectors of edge-sets of paths in a given tree. Edge-Path-Tree graphs have polynomially many cliques as proved in [4] and [7]. Therefore, the problem of finding a clique of maximum weight in these graphs is solvable in strongly polynomial time. In this paper we extend this result to a proper superclass of Edge-Path-Tree graphs. Each graph in the class is defined as the intersection graph of a matrix with no submatrix in a set W of seven small forbidden submatrices. By forbidding an eighth small matrix, our result specializes to Edge-Path-Tree graph