Components of flip graphs of domino tilings in quadriculated cylinder and torus

Let $R$ be a quadriculated surface, possibly with boundary, consisting of unit squares, and each interior vertex being surrounded by 4 squares. A tiling of $R$ is a placement of dominoes (a pair of adjacent squares) so that there are no gaps or overlaps. The flip graph of $R$ is a graph whose vertices are all tilings of $R$ and two tilings are adjacent if we can obtain one from another by a flip (${90}^{\circ }$ rotation of a pair of side-by-side dominoes). By using graph-theoretical approach, we prove that the flip graph of $2m\times (2n+1)$ quadriculated cylinder is still connected, but that of $2m\times (2n+1)$ quadriculated torus consists of two isomorphic components. For a tiling $t$, we associate an integer, called forcing number, as the minimum number of dominoes in $t$ that is contained in no other tilings. As a consequence, we obtain that the forcing numbers of all tilings in $2m\times (2n+1)$ quadriculated cylinder and torus form respectively an integer interval whose maximum value is $(n+1)m$.