The minimum size of a $3$-connected locally nonforesty graph

Abstract

A local subgraph of a graph is the subgraph induced by the neighborhood of a vertex. Thus a graph of order $n$ has $n$ local subgraphs. A graph $G$ is called locally nonforesty if every local subgraph of $G$ contains a cycle. Recently, in studying forest cuts of a graph, Chernyshev, Rauch and Rautenbach posed the conjecture that if $n$ and $m$ are the order and size of a $3$-connected locally nonforesty graph respectively, then $m\ge 7(n-1)/3.$ We solve this problem by determining the minimum size of a $3$-connected locally nonforesty graph of order $n.$ It turns out that the conjecture does not hold.