Cube intersection concepts in median graphs

In this paper, we study different classes of intersection graphs of maximal hypercubes of median graphs. For a median graph G and k ≥ 0 , the intersection graph Q k ( G ) is defined as the graph whose vertices are maximal hypercubes (by inclusion) in G , and two vertices H x and H y in Q k ( G ) are adjacent whenever the intersection H x ∩ H y contains a subgraph isomorphic to Q k . Characterizations of clique-graphs in terms of these intersection concepts when k > 0 , are presented. Furthermore, we introduce the so-called maximal 2-intersection graph of maximal hypercubes of a median graph G , denoted Q m2 ( G ) , whose vertices are maximal hypercubes of G , and two vertices are adjacent if the intersection of the corresponding hypercubes is not a proper subcube of some intersection of two maximal hypercubes. We show that a graph H is diamond-free if and only if there exists a median graph G such that H is isomorphic to Q m2 ( G ) . We also study convergence of median graphs to the one-vertex graph with respect to all these operations.