0 KB411 KB49 KB200820252012Boutin, Debra L.Cockburn, SallyDean, Alice M.Margea, Andrei M.številka:2letnik:5str. 269-288COBISSID:16275801ISSN:1855-3966URN:URN:NBN:SI:doc-F2X1HOEVenDruštvo matematikov, fizikov in astronomov SlovenijeArs mathematica contemporaneadelno urejena množicageometric graphgeometrijski grafgraph theoryhomomorfizemhomomorphismposetteorija grafovPosets of geometric graphs|A geometric graph ?$\overline{G}$? is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call ?$\overline{G}$? a geometric realization of the underlying abstract graph ?$G$?. A geometric homomorphism ?$f: \overline{G} \to \overline{H}$? is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). This work uses geometric homomorphisms to introduce a partial order on the set of isomorphism classes of geometric realizations of an abstract graph ?$G$?. Set ?$\overline{G} \preceq \widehat{G}$? if ?$\overline{G}$? and ?$\widehat{G}$? are geometric realizations of ?$G$? and there is a vertex-injective geometric homomorphism ?$f: \overline{G} \to \widehat{G}$?. This paper develops tools to determine when two geometric realizations are comparable. Further, for ?$3 \le n \le 6$?, this paper provides the isomorphism classes of geometric realizations of ?$P_n$?, ?$C_n$? and ?$K_n$?, as well as the Hasse diagrams of the geometric homomorphism posets (resp., ?$\mathcal{P}_n, \mathcal{C}_n, \mathcal{K}_n$?) of these graphs. The paper also provides the following results for general ?$n$?: each of ?$\mathcal{P}_n$? and ?$\mathcal{C}_n$? has a unique minimal element and a unique maximal element; if ?$k \le n$? then ?$\mathcal{P}_k$? (resp., ?$\mathcal{C}_k$?) is a subposet of ?$\mathcal{P}_n$? (resp., ?$\mathcal{C}_n$?); and ?$\mathcal{K}_n$? contains a chain of length ?$n - 2$?TEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije