Question $30(1$ point$)$

Recall that a node A satisfies strong triadic closure if whenever $\{A,B\}$ and

$\{A,C\}$ are strong $($for $BC),$ there is an edge $\{B,C\}$ that is at least weak

$($and possibly strong$).$ An edge $\{A,B\}$ is a local bridge if in the graph with the

edge $\{A,B\}$ removed, the shortest path from $A$ to $B$ has length at least $3.$

Recall from lecture the following claim and associated proof:

Claim: If A satisfies strong triadic closure and has $2$ strong ties, then any local

bridge involving A must be weak.

Proof.

Let $\{A,B\}$ be a local bridge involving $A.$

WTS $\{A,B\}$ is weak; by way of contradiction, suppose not. That is$,$

suppose that $\{A,B\}$ is strong.

There must be some node $CB$ such that $\{A,C\}$ is a strong tie.

Then by strong triadic closure of $A,$ there must be an edge $\{B,C\}($since

$\{A,B\}$ and $\{A,C\}$ are both strong.$)$