@inproceedings{105b4d8dff374c3e89dba4a0d2ebd8ef,

title = "Drawing graphs using a small number of obstacles",

abstract = "An obstacle representation of a graph G is a set of points in the plane representing the vertices of G, together with a set of polygonal obstacles such that two vertices of G are connected by an edge in G if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number obs(G) of G is the minimum number of obstacles in an obstacle representation of G.We provide the first non-trivial general upper bound on the obsta- cle number of graphs by showing that every nvertex graph G satisfies obs(G) ≤ 2n log n. This refutes a conjecture of Mukkamala, Pach, and P{\'a}lv{\"o}lgyi. For bipartite n-vertex graphs, we improve this bound to n-1. Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound 2Ω(hn) on the number of n-vertex graphs with obstacle number at most h for h < n and an asymptotically matching lower bound Ω(n4/3M 2/3) for the complexity of a collection of M ≥ Ω(n) faces in an arrangement of n2 line segments with 2n endpoints.",

keywords = "Arrangement of line segments, Geometric drawing, Obstacle number, Obstacle representation",

author = "Martin Balko and Josef Cibulka and Pavel Valtr",

note = "Funding Information: The first and the third author acknowledge the support of the project CE-ITI (GA{\v C}R P202/12/G061) of the Czech Science Foundation and the grant GAUK 1262213 of the Grant Agency of Charles University. The first author was also supported by the grant SVV–2015–260223. Part of the research was conducted during the workshop Homonolo 2014 supported by the European Science Foundation as a part of the EuroGIGA collaborative research program (Graphs in Geometry and Algorithms). Publisher Copyright: {\textcopyright} Springer International Publishing Switzerland 2015.; 23rd International Symposium on Graph Drawing and Network Visualization, GD 2015 ; Conference date: 24-09-2015 Through 26-09-2015",

year = "2015",

month = jan,

day = "1",

doi = "10.1007/978-3-319-27261-0_30",

language = "English",

isbn = "9783319272603",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "360--372",

editor = "{Di Giacomo}, Emilio and Anna Lubiw",

booktitle = "Graph Drawing and Network Visualization - 23rd International Symposium, GD 2015, Revised Selected Papers",

address = "Germany",

}