Block Graph of a Graph

Kelkar, A. , Jaysurya, K. , Nagesh, H. M.
Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 1.

Abstract:
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.

Keywords: rossing number, inner vertex number, dutch windmill graph, complete graph.

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