Cuts on a Circle - Math Problems

For any three distinct vertices,

c(u,w)\ge \min\{c(u,v),c(v,w)\}.

Indeed, removing edges to disconnect $u$ from $w$ must also disconnect either $u$ from $v$ or $v$ from $w$ . Otherwise, the remaining paths through $v$ would still connect $u$ to $w$ .

For each positive integer $t$ , put two distinct vertices in the same class exactly when $c(u,v)\ge t$ . The inequality makes this relation transitive, so it defines a partition.

As $t$ increases, classes can split but cannot merge. Consequently, any two classes appearing at any thresholds are either disjoint or one contains the other.

Let $M=\max_{u\ne v}c(u,v)$ . At threshold $1$ , all vertices belong to one class. At threshold $M+1$ , every class is a singleton.

For any three distinct vertices,

c(u,w)\ge \min\{c(u,v),c(v,w)\}.

Indeed, removing edges to disconnect $u$ from $w$ must also disconnect either $u$ from $v$ or $v$ from $w$ . Otherwise, the remaining paths through $v$ would still connect $u$ to $w$ .

For each positive integer $t$ , put two distinct vertices in the same class exactly when $c(u,v)\ge t$ . The inequality makes this relation transitive, so it defines a partition.

As $t$ increases, classes can split but cannot merge. Consequently, any two classes appearing at any thresholds are either disjoint or one contains the other.

Let $M=\max_{u\ne v}c(u,v)$ . At threshold $1$ , all vertices belong to one class. At threshold $M+1$ , every class is a singleton.

Nested connectivity classes

Nested connectivity classes