TY - JOUR

T1 - BOUNDING AND COMPUTING OBSTACLE NUMBERS OF GRAPHS

AU - Balko, Martin

AU - Chaplick, Steven

AU - Ganian, Robert

AU - Gupta, Siddharth

AU - Hoffmann, Michael

AU - Valtr, Pavel

AU - Wolff, Alexander

N1 - Publisher Copyright:
© 2024 Society for Industrial and Applied Mathematics Publications. All rights reserved.

PY - 2024/6/1

Y1 - 2024/6/1

N2 - An obstacle representation of a graph G consists of a set of pairwise disjoint simply connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each n-vertex graph is O(nlogn) [M. Balko, J. Cibulka, and P. Valtr, Discrete Comput. Geom., 59 (2018), pp. 143-164] and that there are n-vertex graphs whose obstacle number is \Omega(n/(loglogn)2) [V. Dujmovi\'c and P. Morin, Electron. J. Combin., 22 (2015), 3.1]. We improve this lower bound to \Omega(n/loglogn) for simple polygons and to \Omega(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmovi\'c and Morin. We also show that if the drawing of some n-vertex graph is given as part of the input, then for some drawings \Omega(n2) obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph G is fixed-parameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NP-hard to decide whether G admits an obstacle representation using P as the only obstacle.

AB - An obstacle representation of a graph G consists of a set of pairwise disjoint simply connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each n-vertex graph is O(nlogn) [M. Balko, J. Cibulka, and P. Valtr, Discrete Comput. Geom., 59 (2018), pp. 143-164] and that there are n-vertex graphs whose obstacle number is \Omega(n/(loglogn)2) [V. Dujmovi\'c and P. Morin, Electron. J. Combin., 22 (2015), 3.1]. We improve this lower bound to \Omega(n/loglogn) for simple polygons and to \Omega(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmovi\'c and Morin. We also show that if the drawing of some n-vertex graph is given as part of the input, then for some drawings \Omega(n2) obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph G is fixed-parameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NP-hard to decide whether G admits an obstacle representation using P as the only obstacle.

KW - convex obstacle number

KW - obstacle number

KW - visibility

UR - http://www.scopus.com/inward/record.url?scp=85195195630&partnerID=8YFLogxK

U2 - 10.1137/23M1585088

DO - 10.1137/23M1585088

M3 - Article

AN - SCOPUS:85195195630

SN - 0895-4801

VL - 38

SP - 1537

EP - 1565

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 2

ER -