Sectional split extensions arising from lifts of groups

Covering techniques have recently emerged as an effective tool used for classification of several infinite families of connected symmetric graphs. One commonly encountered technique is based on the concept of lifting groups of automorphisms along regular covering projections ?$\wp \colon \tilde{X} \to X$?. Efficient computational methods are known for regular covers with cyclic or elementary abelian group of covering transformations CT?$(\wp)$?. In this paper we consider the lifting problem with an additional condition on how a group should lift: given a connected graph ?$X$? and a group ?$G$? of its automorphisms, find all connected regular covering projections ?$\wp \colon \tilde{X} \to X$? along which ?$G$? lifts as a sectional split extension. By this we mean that there exists a complement ?$\overline{G}$? of CT?$(\wp)$? within the lifted group ?$\tilde{G}$? such that ?$\overline{G}$? has an orbit intersecting each fibre in at most one vertex. As an application, all connected elementary abelian regular coverings of the complete graph ?$K_4$? along which a cyclic group of order 4 lifts as a sectional split extension are constructed. Krovne tehnike so se izkazale kot učinkovito orodje pri klasifikaciji več neskončnih družin povezanih simetričnih grafov. Ena izmed pogostih tehnik, s katerimi se srečamo, temelji na konceptu dviga avtomorfizmov grup vzdolž regularnih krovnih projekcij ?$\wp \colon \tilde{X} \to X$?. Učinkovite računske metode so znane v primeru regulanih krovov s ciklično ali elementarno abelsko grupo krovnih transformacij CT?$(\wp)$?. V članku študiramo problem dviga pri dodatnem pogoju, kako naj se grupa dvigne: za dani povezan graf? $X$? in podgrupo ?$G$? njegovih avtomorfizmov poišči vse povezane regularne krovne projekcije ?$\wp \colon \tilde{X} \to X$?, vzdolž katerih se ?$G$? dvigne kot sekcijska razcepna razširitev. To pomeni, da obstajakomplement ?$\overline{G}$? k CT?$(\wp)$? znotraj dvignjene grupe ?$\tilde{G}$?, tako da ima ?$\overline{G}$? orbito, ki seka vsako vlakno v največ enem vozlišču. Za ilustracijo konstruiramo vse povezane elementarno ableske regularne krove polnega grafa ?$K_4$?, vzdolž katerih se ciklična grupa reda 4 dvigne kot sekcijska razcepna razširitev.