Almost minimal models of log surfaces

We generalize Miyanishi's theory of almost minimal models of log smooth surfaces with reduced boundary to the case of arbitrary log surfaces defined over an algebraically closed field. Given an MMP run of a log surface $(X,D)$ we define and construct its almost minimal model, whose underlying surface has singularities not worse than $X$ and which differs from a minimal model by a contraction of some curves supported in the boundary only. For boundaries of type $rD$, where $D$ is reduced and $r\in [0,1]\cap \mathbb{Q}$, we show that if $X$ is smooth or $r\in [0,\frac{1}{2}]$ then the construction respects $(1-r)$-divisorial log terminality and $(1-r)$-log canonicity. We show that the assumptions are optimal, too.