Geodesic
Block graph of a graph
Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 1, pp. 74-78

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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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     title = {Block graph of a graph},
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A. Kelkar; K. Jaysurya; H. M. Nagesh. Block graph of a graph. Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 1, pp. 74-78. http://geodesic.mathdoc.fr/item/VMJ_2019_21_1_a6/