For a connected bipartite graph $G$ are the two following properties equivalent:

1. $G$ is chordal bipartite

2. For all triplet of vertices $\{ v_1, v_2, v_3 \}$ in $G$ there exist three geodesics from $v_1$ to $v_2$, $v_1$ to $v_3$ and $v_2$ to $v_3$ respectively, whose union form a subtree of $G$.

If yes, is there any reference for this? If no what would be a counter example?

Thank you for the answer

For a connected bipartite graph $G$ are the two following properties equivalent:

1)Every minimal cycle in $G$ has length 4, that is every cycle of length strictly greater than 4 can be divided in cycles of length 4 ($Z^d$ is such a graph).

2. For all triplet of vertices $\{ v_1, v_2, v_3 \}$ in $G$ there exist three geodesics from $v_1$ to $v_2$, $v_1$ to $v_3$ and $v_2$ to $v_3$ respectively, whose union form a subtree of $G$.

If yes, is there any reference for this? If no what would be a counter example?

Thank you for the answer

For a bipartite graph $G$ are the two following properties equivalent:

1. $G$ is chordal bipartite

2. For all triplet of vertices $\{ v_1, v_2, v_3 \}$ in $G$ there exist three geodesics from $v_1$ to $v_2$, $v_1$ to $v_3$ and $v_2$ to $v_3$ respectively, whose union form a subtree of $G$.

If yes, is there any reference for this? If no what would be a counter example?

Thank you for the answer

For a connected bipartite graph $G$ are the two following properties equivalent:

1. $G$ is chordal bipartite

2. For all triplet of vertices $\{ v_1, v_2, v_3 \}$ in $G$ there exist three geodesics from $v_1$ to $v_2$, $v_1$ to $v_3$ and $v_2$ to $v_3$ respectively, whose union form a subtree of $G$.

If yes, is there any reference for this? If no what would be a counter example?

Thank you for the answer