5 research outputs found
Lower bounds on the obstacle number of graphs
Given a graph , an {\em obstacle representation} of is a set of points
in the plane representing the vertices of , together with a set of connected
obstacles such that two vertices of are joined by an edge if and only if
the corresponding points can be connected by a segment which avoids all
obstacles. The {\em obstacle number} of is the minimum number of obstacles
in an obstacle representation of . It is shown that there are graphs on
vertices with obstacle number at least
Drawing Cubic Graphs with the Four Basic Slopes
We show that every cubic graph can be drawn in the plane with straight-line edges using only the four basic slopes, {0, π/4, π/2, 3π/4}. We also prove that four slopes have this property if and only if we can draw K4 with them
Asymptotically optimal pairing strategy for Tic-Tac-Toe with numerous directions
We show that there is an m = 2n + o(n), such that, in the Maker-Breaker game played on Zd where Maker needs to put at least m of his marks consecutively in one of n given winning directions, Breaker can force a draw using a pairing strategy. This improves the result of Kruczek and Sundberg [15] who showed that such a pairing strategy exits if m ≥ 3n. A simple argument shows that m has to be at least 2n+1 if Breaker is only allowed to use a pairing strategy, thus the main term of our bound is optimal.
Obstacles, slopes and tic-tac-toe: an excursion in discrete geometry and combinatorial game theory
The minimum number of slopes used in a straight-line drawing of G is called the slope number of G. We show that every cubic graph can be drawn in the plane with straight line edges using only the four basic slopes {0, π/4, π/2,−π/4}. We also prove that four slopes have this property if and only if we can draw K4 with them. Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of obstacles (connected polygons)
such that two vertices of G are joined by an edge if and only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of G is the minimum number of obstacles in an obstacle representation of G. We show that there are graphs on n vertices with obstacle number (n/log n). We show that there is an m = 2n + o(n), such that, in the Maker-Breaker game played on Zd where Maker needs to put at least m of his marks consecutively in one
of n given winning directions, Breaker can force a draw using a pairing strategy. This improves the result of Kruczek and Sundberg who showed that such a pairing strategy exits if m ≥ 3n. A simple argument shows that m has to be at least 2n+1 if Breaker is only allowed to use a pairing strategy, thus the main term of our bound is optimal.Ph. D.Includes bibliographical referencesIncludes vitaby V S Padmini Mukkamal