Geodesic
Simultaneous Embeddings with Few Bends and Crossings
Journal of Graph Algorithms and Applications, Tome 23 (2019) no. 4, pp. 683-713.

Voir la notice de l'article provenant de la source Journal of Graph Algorythms and Applications website

A simultaneous embedding with fixed edges ($\rm{S{\small EFE}}$) of two planar graphs $R$ and $B$ is a pair of plane drawings of $R$ and $B$ that coincide when restricted to the common vertices and edges of $R$ and $B$. We show that whenever $R$ and $B$ admit a $\rm{S{\small EFE}}$, they also admit a $\rm{S{\small EFE}}$ in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically: if $R$ and $B$ are trees then one bend per edge and four crossings per edge pair suffice (and one bend per edge is sometimes necessary), if $R$ is a planar graph and $B$ is a tree then six bends per edge and eight crossings per edge pair suffice, and if $R$ and $B$ are planar graphs then six bends per edge and sixteen crossings per edge pair suffice. Our results simultaneously improve on a paper by Grilli et al. (GD'14), which proves that nine bends per edge suffice, and on a paper by Chan et al. (JGAA '15), which proves that twenty-four crossings per edge pair suffice. 
@article{JGAA_2019_23_4_a3,
     author = {Fabrizio Frati and Michael Hoffmann and Vincent Kusters},
     title = {Simultaneous {Embeddings} with {Few} {Bends} and {Crossings}},
     journal = {Journal of Graph Algorithms and Applications},
     pages = {683--713},
     publisher = {mathdoc},
     volume = {23},
     number = {4},
     year = {2019},
     doi = {10.7155/jgaa.00507},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00507/}
}
TY  - JOUR
AU  - Fabrizio Frati
AU  - Michael Hoffmann
AU  - Vincent Kusters
TI  - Simultaneous Embeddings with Few Bends and Crossings
JO  - Journal of Graph Algorithms and Applications
PY  - 2019
SP  - 683
EP  - 713
VL  - 23
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00507/
DO  - 10.7155/jgaa.00507
LA  - en
ID  - JGAA_2019_23_4_a3
ER  - 
%0 Journal Article
%A Fabrizio Frati
%A Michael Hoffmann
%A Vincent Kusters
%T Simultaneous Embeddings with Few Bends and Crossings
%J Journal of Graph Algorithms and Applications
%D 2019
%P 683-713
%V 23
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00507/
%R 10.7155/jgaa.00507
%G en
%F JGAA_2019_23_4_a3
Fabrizio Frati; Michael Hoffmann; Vincent Kusters. Simultaneous Embeddings with Few Bends and Crossings. Journal of Graph Algorithms and Applications, Tome 23 (2019) no. 4, pp. 683-713. doi : 10.7155/jgaa.00507. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00507/

Cité par Sources :