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.
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