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In a region $R$ consisting of unit squares, a domino is the union of twoadjacent squares and a (domino) tiling is a collection of dominoes withdisjoint interior whose union is the region. The flip graph $\mathcal{T}(R)$ isdefined on the set of all tilings of $R$ such that two tilings are adjacent ifwe change one to another by a flip (a $90^{\circ}$ rotation of a pair ofside-by-side dominoes). It is well-known that $\mathcal{T}(R)$ is connectedwhen $R$ is simply connected. By using graph theoretical approach, we show thatthe flip graph of $2m\times(2n+1)$ quadriculated cylinder is still connected,but the flip graph of $2m\times(2n+1)$ quadriculated torus is disconnected andconsists of exactly two isomorphic components. For a tiling $t$, we associatean integer $f(t)$, forcing number, as the minimum number of dominoes in $t$that is contained in no other tilings. As an application, we obtain that theforcing numbers of all tilings in $2m\times (2n+1)$ quadriculated cylinder andtorus form respectively an integer interval whose maximum value is $(n+1)m$.

Components of domino tilings under flips in quadriculated cylinder and torus