22 research outputs found
Universal Geometric Graphs
We introduce and study the problem of constructing geometric graphs that have
few vertices and edges and that are universal for planar graphs or for some
sub-class of planar graphs; a geometric graph is \emph{universal} for a class
of planar graphs if it contains an embedding, i.e., a
crossing-free drawing, of every graph in .
Our main result is that there exists a geometric graph with vertices and
edges that is universal for -vertex forests; this extends to
the geometric setting a well-known graph-theoretic result by Chung and Graham,
which states that there exists an -vertex graph with edges
that contains every -vertex forest as a subgraph. Our bound on
the number of edges cannot be improved, even if more than vertices are
allowed.
We also prove that, for every positive integer , every -vertex convex
geometric graph that is universal for -vertex outerplanar graphs has a
near-quadratic number of edges, namely ; this almost
matches the trivial upper bound given by the -vertex complete
convex geometric graph.
Finally, we prove that there exists an -vertex convex geometric graph with
vertices and edges that is universal for -vertex
caterpillars.Comment: 20 pages, 8 figures; a 12-page extended abstracts of this paper will
appear in the Proceedings of the 46th Workshop on Graph-Theoretic Concepts in
Computer Science (WG 2020
Area-Efficient Drawings of Outer-1-Planar Graphs
We study area-efficient drawings of planar graphs: embeddings of graphs on an integer grid so that the bounding box of the drawing is minimized. Our focus is on the class of outer-1-planar graphs: the family of planar graphs that can be drawn on the plane with all vertices on the outer-face so that each edge is crossed at most once. We first present two straight-line drawing algorithms that yield small-area straight-line drawings for the subclass of complete outer-1-planar graphs. Further, we give an algorithm that produces an orthogonal drawing of any outer-1-plane graph in O(n log n) area while keeping the number of bends per edge relatively small
Tree Drawings Revisited
We make progress on a number of open problems concerning the area requirement for drawing trees on a grid. We prove that
1) every tree of size n (with arbitrarily large degree) has a straight-line drawing with area n2^{O(sqrt{log log n log log log n})}, improving the longstanding O(n log n) bound;
2) every tree of size n (with arbitrarily large degree) has a straight-line upward drawing with area n sqrt{log n}(log log n)^{O(1)}, improving the longstanding O(n log n) bound;
3) every binary tree of size n has a straight-line orthogonal drawing with area n2^{O(log^*n)}, improving the previous O(n log log n) bound by Shin, Kim, and Chwa (1996) and Chan, Goodrich, Kosaraju, and Tamassia (1996);
4) every binary tree of size n has a straight-line order-preserving drawing with area n2^{O(log^*n)}, improving the previous O(n log log n) bound by Garg and Rusu (2003);
5) every binary tree of size n has a straight-line orthogonal order-preserving drawing with area n2^{O(sqrt{log n})}, improving the O(n^{3/2}) previous bound by Frati (2007)
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum