3D Visibility Representations of 1-planar Graphs

We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of G.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

On Alternating Products of Graph Relations

It is well-known that one can give an elegant version of the Kuratowskitype theorem for the projective plane by means of the five elementary relations R i ; i = 0; 1; : : : ; 4; on the set \Gamma of all finite, undirected graphs without loops and multiple edges. Furthermore, these five relations play an interesting role in didactics of mathematics. Following a theory given in [2], C.Thies investigates them in [3]. In order to show that R 0 ; R 1 ; : : : ; R 4 are an appropriate curriculum he has to deal with so-called alternating products RT (R i ) ffi RT (R \Gamma1 i ) ffi RT (R i ) ffi RT (R \Gamma1 i ) ffi : : : or RT (R \Gamma1 i ) ffi RT (R i ) ffi RT (R \Gamma1 i ) ffi RT (R i ) ffi : : : ; i = 0; 1; : : : ; 4; where RT (R i ) denotes the reflexive transitive closure of R i and RT (R \Gamma1 i ) = RT (R i ) \Gamma1 the reflexive, transitive closure of R \Gamma1 i . Here, it is shown that, in case of i = 0, there exists exactly one alternating product in the set ..

On Alternating Products of Graph Relations

It is well-known that one can give an elegant version of the Kuratowskitype theorem for the projective plane by means of the five elementary relations R i ; i = 0; 1; : : : ; 4; on the set \Gamma of all finite, undirected graphs without loops and multiple edges. Furthermore, these five relations play an interesting role in didactics of mathematics. Following a theory given in [2], C.Thies investigates them in [3]. In order to show that R 0 ; R 1 ; : : : ; R 4 are an appropriate curriculum he has to deal with so-called alternating products RT (R i ) ffi RT (R \Gamma1 i ) ffi RT (R i ) ffi RT (R \Gamma1 i ) ffi : : : or RT (R \Gamma1 i ) ffi RT (R i ) ffi RT (R \Gamma1 i ) ffi RT (R i ) ffi : : : ; i = 0; 1; : : : ; 4; where RT (R i ) denotes the reflexive transitive closure of R i and RT (R \Gamma1 i ) = RT (R i ) \Gamma1 the reflexive, transitive closure of R \Gamma1 i . Here, it is shown that, in case of i = 0, there exists exactly one alternating product in the set ..

(a,d)-edge-antimagic total labelings of caterpillars

For a graph G = (V,E), a bijection g from V (G)∪E(G) into {1, 2, ..., |V (G)|+|E(G)|} is called (a, d)-edge-antimagic total labeling of G if the edge-weights w(xy) = g(x) + g(y) + g(xy), xy ∈ E(G), form an arithmetic progression with initial term a and common difference d. An (a, d)-edge-antimagic total labeling g is called super (a, d)-edge-antimagic total if g(V (G)) = {1, 2, ..., |V (G)|}. We study super (a, d)-edge-antimagic total properties of stars Sn and caterpillar Sn1,n2,...,nr .C