356 KB41 KB200820252017Gerbner, DánielKeszegh, BalázsPálvölgyi, DömötörRote, GünterWiener, Gáborletnik:12številka:2str. 301-314COBISSID:18165081ISSN:1855-3966URN:URN:NBN:SI:doc-80QO6PXUenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneagrafiSearch for the end of a path in the d-dimensional grid and in other graphs|We consider the worst-case query complexity of some variants of certain PPAD- complete search problems. Suppose we are given a graph ?$G$? and a vertex ?$s \in V (G)$?. We denote the directed graph obtained from ?$G$? by directing all edges in both directions by ?$G'$?. ?$D$? is a directed subgraph of ?$G'$? which is unknown to us, except that it consists of vertex-disjoint directed paths and cycles and one of the paths originates in ?$s$?. Our goal is to find an endvertex of a path by using as few queries as possible. A query specifies a vertex ?$v \in V (G)$?, and the answer is the set of the edges of ?$D$? incident to ?$v$?, together with their directions. We also show lower bounds for the special case when ?$D$? consists of a single path. Our proofs use the theory of graph separators. Finally, we consider the case when the graph ?$G$? is a grid graph. In this case, using the connection with separators, we give asymptotically tight bounds as a function of the size of the grid, if the dimension of the grid is considered as fixed. In order to do this, we prove a separator theorem about grid graphs, which is interesting on its own rightObravnavamo najslabši primer iskalne kompleksnosti nekaterih različic določenih PPAD-polnih iskalnih problemov. Denimo da je dan graf ?$G$? in vozlišče ?$s \in V (G)$?. Označimo usmerjeni graf dobljen iz grafa ?$G$? z usmeritvijo vseh povezav v obe smeri z ?$G'$?. ?$D$? je usmerjen podgraf grafa ?$G'$?, ki nam je neznan, vemo le, da sestoji iz vozliščno-disjunktnih usmerjenih poti in ciklov, in da se ena od poti začne v ?$s$?. Naš cilj je najti končno vozlišče poti, za to pa uporabiti kar se malo iskanj. Iskanje specificira vozlišče ?$v \in V (G)$?, in odgovor je množica povezav grafa ?$D$? incidentnih vozlišču ?$v$?, skupaj z njihovimi smermi. Določimo tudi spodnje meje za posebni primer, ko ?$D$? sestoji iz ene same poti. Naši dokazi uporabljajo teorijo grafovskih ločevalcev. Nazadnje obravnavamo primer, ko je graf ?$G$? rešetkast graf. V tem primeru, uporabljajoč zvezo z ločevalci, podamo asimptotsko tesne meje, izražene z velikostjo rešetke, če je dimenzija rešetke smatrana za fiksno. V ta namen dokažemo ločevalske izreke v zvezi z rešetkastimi grafi, ki so zanimivi tudi sami po sebiTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije