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If G is a $k$-connnected graph and $u,v_1,v_2, \dots , v_t$ are $t+1$ distinct vertices in $G$ for $2 \leq t \leq k$, $G$ contains a $u-v_i$ path for each $1 \leq i \leq t$, every two paths of which have only $u$ in common.

I believe the result comes from Whitney's theorem but how to show this?

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We construct a new graph $G'$ by adding a new vertex $w$ and adding $t$ new edges $(w, v_1)$, $(w, v_2)$, $\cdots$, and $(w, v_t)$ to $G$. The resulting graph $G'$ is $t$-connected. By the Menger's theorem, there exist $t$ vertex-disjoint paths from $u$ to $w$ in $G'$. Note any path from $u$ to $w$ must pass one of $\{v_1, \cdots, v_t\}$. Therefore, the $t$ vertex-disjoint paths from $u$ to $w$ correspond to $t$-vertex disjoint paths from $u$ to $\{v_1, \cdots, v_t\}$ respectively.

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  • $\begingroup$ Do you mean after adding vertex $w$ then $t=k$ so there exists $t$ internally disjoint paths ? isn't that whiteny's theorem which can be provedby menger's theorem ? $\endgroup$ – Frank Mar 17 '17 at 17:21
  • $\begingroup$ ok.Why can we suppose the existence of $w$? $\endgroup$ – Frank Mar 17 '17 at 17:36
  • $\begingroup$ what if I change the claim to If G is a $k$-edge connected graph and $v,v_1,v_2, \dots , v_k$ are $k+1$ vertices of $G$,then for $i$=$1,2...,k$, there exists a $v-v_i$ path $P_i$ such that each path $P_i$ contains exactly one vertex ${v_1,v_2,...v_k}$ and for $i$ doesn't equal to$ j$ .$P_i$ and $P_j$ are edge disjoint .Is it valid ? $\endgroup$ – Frank Mar 18 '17 at 8:22
  • $\begingroup$ How would you suggest to do that? $\endgroup$ – Frank Mar 18 '17 at 9:03

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