# Byzantine Generals Problem - Regular Set of Neighbors

Going through Lamport's paper, I'm a little confused by the second part of the regular set of neighbors definition.

1. A set of nodes $$\{i_1, \ldots, i_p\}$$ is said to be a regular set of neighbors of a node if :
1. each $$i_j$$ is a neighbor of $$i$$, and
2. for any general $$k$$ different from $$i$$, there exist paths $$y_j,_k$$ from $$i_j$$ to $$k$$ not passing through $$i$$ such that any two different paths $$y_i,_k$$ have no node in common other than $$k$$.
2. The graph $$G$$ is said to be $$p$$-regular if every node has a regular set of neighbors consisting of $$p$$ distinct nodes.

I'm assuming $$i_j$$ is a node in the set. Is it right to assume $$k$$ is any other node not $$i$$ or any node not $$i$$ and not in the set? Also, is it saying that every $$y_j,_k$$ path will not have a common node or that there must be some $$y_j,_k$$ paths that don't share a common node?

I put the example graph from Lamport's paper with some labels on the nodes (that was edited with a track pad. Sorry!) just in case it'll help give an explaination. Any and all help is much appreciated!

• (1) $k$ is any node different from $i$. Not necessarily in $\{i_1,\ldots,i_p\}$. (2) For every $1 \leq j_1 < j_2 \leq p$, the paths $y_{j_1,k},y_{j_2,k}$ intersect only in $k$. Oct 17 '19 at 20:44