# Connection between connectivity and cohesion of a graph

Tutte [1] proved that, for every $3$-connected graph $G$ and vertices $u$ and $v$, there exists a nonseparating $uv$-path.

A graph $G$ is $t$-cohesive if $G$ is connected, has at least two vertices, and for every pair of distinct vertices, $u$ and $v$, $d(u)+d(v)+d(u,v)\geq t$, where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance from $u$ to $v$. Locke et al. [2] show that every $2k$-cohesive graph contains a non-separating path on $k$ vertices.

My question: what is the connection between connectivity and cohesion of a graph? Any existing results about the condition of non-separating path on $k$ vertices (except the above mentioned one)? Especially the condition which combines connectivity and cohesion together. (In the existing literature, there are some results of cohesive structure in social networks, which is different from the definition above.)

[1] Tutte, W. T.: How to draw a graph, Proc. London Math. Soc., 13 (1963), 743–767.

[2] Locke, Stephen C., Phil Tracy, and H-J. Voss. "Highly Cohesive Graphs Have Long Nonseparating Paths: 10647." American Mathematical Monthly (2001): 470-472.