Minimal biconnected graphs

D. V. Karpov

Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VI, Tome 417 (2013), pp. 106-127

Résumé

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 $G_T$.