We construct in Logspace non-zero circulations for H-minor free graphs where H is any fixed graph with a single crossing. This extends the list of planar, bounded genus and K_{3,3}-free, K_5-free graphs for which non-zero circulations are already known. This implies static bounds of UL and SPL, respectively, for the problems of reachability and maximum matching in this class of graphs. Further, we prove the following dynamic complexity bounds for classes of graphs where polynomially bounded, skew-symmetric non-zero circulations are in Logspace: (1) A DynFO[Parity] bound (under polylogarithmic changes per step) for constructing perfect matching and maximum matching. (2) A DynFO[Parity] bound (under polylogarithmic changes per step) for distance. (3) A DynFO bound (under slightly sublogarithmic changes per step) for maximum cardinality matching size. The main technical contributions include the construction of polynomially bounded non-zero circulation weights in single crossing minor free graphs based on existing results [AGGT16,DKMTVZ20]; using the approach from [FGT16] to construct dynamic isolating weights and generalizing the rank algorithm from [DKMSZ15,DKMSZ18] to handle slightly sublogarithmic many (O(log n/log log n)) changes.
翻译:我们用Logspace非零环流为H-minor免费图表建造H-minor 自由环流,H是任何固定的图形,有单一交叉点的H是任何固定的图形。这扩展了平面图列表,有捆绑的genus 和 K ⁇ 3,3}无,无K_5无,没有零环流已经已知。这意味着UL 和 SPL 的静态边框,分别是本类图形的可达性和最大匹配问题。此外,我们证明,在Logspace中,H是任何固定的图形,H是任何固定的。这扩展了平面图的列表:(1) DynFo[Paite] 捆绑(不设于多轨变化中),用于构建完美匹配和最大匹配。 (3) DynFO绑定(在每步略小次小的子对数变化下),用于最大基底大小的图形。主要技术贡献包括:在Logspacespace空间中,从单交叉的Sy-Z平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面,[AGTGTGTGTGTGTGTGTGTGTGMS+平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平