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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 2013 Volume 417, Pages 106–127 (Mi znsl5707)

This article is cited in 10 papers

Minimal biconnected graphs

D. V. Karpovab

a St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia
b St. Petersburg State University, St. Petersburg, Russia

Abstract: A biconnected graph is called minimal, if it becomes not biconnected after deleting any edge. We consider minimal biconnected graphs that have minimal number of vertices of degree 2. Denote the set of all such graphs on $n$ vertices by $\mathcal GM(n)$. It is known that a graph from $\mathcal GM(n)$ contains exactly $\lceil\frac{n+4}3\rceil$ vertices of degree 2. We prove that for $k\ge1$ the set $\mathcal GM(3k+2)$ consists of all graphs of type $G_T$, where $T$ is a tree on $k$ vertices which vertex degrees do not exceed 3. The graph $G_T$ is constructed of two copies of the tree $T$: to each pair of correspondent vertices of these two copies that have degree $j$ in $T$ we add $3-j$ new vertices of degree 2 adjacent to this pair. Graphs of the sets $\mathcal GM(3k)$ and $\mathcal GM(3k+1)$ are described with the help of graphs $G_T$.

Key words and phrases: connectivity, biconnected graph, decomposition, blocks.

UDC: 519.173.1

Received: 05.11.2013


 English version:
Journal of Mathematical Sciences (New York), 2015, 204:2, 244–257

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