Netlike partial cubes III. The median cycle property

A graph G has the Median Cycle Property (MCP) if every triple ( u 0 , u 1 , u 2 ) of vertices of G admits a unique median or a unique median cycle, that is a gated cycle C of G such that for all i , j , k ∈ { 0 , 1 , 2 } , if x i is the gate of u i in C , then: { x i , x j } ⊆ I G ( u i , u j ) if i ≠ j , and d G ( x i , x j ) < d G ( x i , x k ) + d G ( x k , x j ) . We prove that a netlike partial cube has the MCP if and only if it contains no triple of convex cycles pairwise having an edge in common and intersecting in a single vertex. Moreover a finite netlike partial cube G has the MCP if and only if G can be obtained from a set of even cycles and hypercubes by successive gated amalgamations, and equivalently, if and only if G can be obtained from K 1 by a sequence of special expansions. We also show that the geodesic interval space of a netlike partial cube having the MCP is a Pash–Peano space (i.e. a closed join space).