Block graph of a graph

The block graph of a graph G, written B(G), is the graph whose vertices are the blocks of G and in which two vertices are adjacent whenever the corresponding blocks have a cut-vertex in common. We study the properties of B(G) and present the characterization of graphs whose B(G) are planar, outerplanar, maximal outerplanar, minimally non-outerplanar, Eulerian, and Hamiltonian. A necessary and sufficient condition for B(G) to have crossing number one is also presented.