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

TL;DR

This paper determines the smallest number of edges in a $k$-connected graph where every local neighborhood contains a cycle, focusing on graphs that are locally nonforesty.

Contribution

It establishes the minimum size of $k$-connected locally nonforesty graphs of a given order, a new result in graph theory.

Findings

Derived the minimum number of edges for such graphs.

Characterized the structure of $k$-connected locally nonforesty graphs.

Provided bounds and exact values for specific parameters.

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. Clearly, a graph is locally nonforesty if and only if every vertex of the graph is the hub of a wheel. We determine the minimum size of a $k$-connected locally nonforesty graph of order $n.$