Matching Edges and Faces in Polygonal Partitions

AbstractWe define general Laman (count) conditions for edges and faces of polygonal partitions in the plane. Several well-known classes, including k-regular partitions, k-angulations, and rank-k pseudo-triangulations, are shown to fulfill such conditions. As an implication, non-trivial perfect matchings exist between the edge sets (or face sets) of two such structures when they live on the same point set. We also describe a link to spanning tree decompositions that applies to quadrangulations and certain pseudo-triangulations

Population genetic structures of two New Zealand corophiid amphipods and the presence of morphologically cryptic species: Implications for the conservation of diversity

We evaluated the population genetic structures (allozyme variation) of Chaetocorophium lucas i (Hurley 1954) Karaman 1979 and Paracoro‐phium excavatum (Thomson 1884) Stebbing 1899 (Amphipoda: Crustacea) from the North Island and South Island of New Zealand. Individuals of C. lucasi were found in both freshwater and estuarine habitats, whereas P. excavatum was restricted to estuarine sites. Allele frequencies showed significant inter‐ and intra‐specific differences among populations of both C. lucasi and P. excavatum. C. lucasi appears to be a single species, albeit with considerable genetic differentiation among populations (FSX = 0.51). However, P. excavatum revealed even greater levels of genetic differentiation among populations (FST = 0.81), and may consist of at least three genetically distinct species. Geographic distance, perhaps dictated by prevailing ocean currents and overland dispersal routes, best explained observed patterns of differentiation among populations of both species. From a conservation perspective, the presence of morphologically cryptic species suggests that current taxonomic inventories of aquatic amphipods may be grossly underestimated