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 Qk(G) is defined as the graph whose vertices are maximal hypercubes (by inclusion) in G, and two vertices Hx and Hy in Qk(G) are adjacent whenever the intersection Hx ∩ Hy contains a subgraph isomorphic to Qk. Characterizations of clique-graphs in terms of these intersection concepts when k > 0, are presented. Furthermore, we introduce the socalled maximal 2-intersection graph of maximal hypercubes of a median graph G, denoted Qm2(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 Qm2(G). We also study convergence of median graphs to the one-vertex graph with respect to all these operations. © 2008 Elsevier B.V. All rights reserved.