@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 = "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",

}