Geodesic
On the Numbers of Cut-Vertices and End-Blocks in 4-Regular Graphs
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 1, pp. 127-136

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A cut-vertex in a graph G is a vertex whose removal increases the number of connected components of G. An end-block of G is a block with a single cut-vertex. In this paper we establish upper bounds on the numbers of end-blocks and cut-vertices in a 4-regular graph G and claw-free 4-regular graphs. We characterize the extremal graphs achieving these bounds.
Keywords: 4-regular graph, claw-free, cut-vertices, end-blocks 
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Wang, Dingguo; Shan, Erfang. On the Numbers of Cut-Vertices and End-Blocks in 4-Regular Graphs. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 1, pp. 127-136. http://geodesic.mathdoc.fr/item/DMGT_2014_34_1_a10/