Angle-monotonicity of theta-graphs for points in convex position

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Bakhshesh D.; Farshi M.

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Journal Paper

Paper Information

Journal: Scientia Iranica Year:2023 | Volume:30 | Issue:Transactions on Computer Science & Engineering and Electrical Engineering (D)6 Page(s): 2116-2123

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Information Journal Paper

Title

Angle-monotonicity of theta-graphs for points in convex position

Author(s)

Bakhshesh D. | Farshi M. | Issue Writer Certificate

Keywords

Angle-monotone path

Theta-graph

Stretch factor

Convex position

Abstract

For $0<\gamma<180^{\circ}$, a geometric path $P=(p_1,\ldots, p_n)$ is called angle-monotone with width $\gamma$ from $p_1$ to $p_n$ if there exists a closed wedge of angle $\gamma$ such that every directed edge $\overrightarrow{p_ip_{i+1}}$ of~$P$ lies inside the wedge whose apex is $p_i$. A geometric graph $G$ is called angle-monotone with width $\gamma$ if for any two vertices $p$ and $q$ in $G$, there exists an Angle-monotone path with width $\gamma$ from $p$ to $q$. In this paper, we show that for any integer $k\geq 1$ and any $i\in\{2, 3, 4, 5\}$, the Theta-graph $\Theta_{4k+i}$ on a set of points in Convex position is angle-monotone with width $90^\circ+\frac{i\theta}{4}$, where $\theta=\frac{360^{\circ}}{4k+i}$. Moreover, we present two sets of points in the plane, one in Convex position and the other in non-convex position, to show that for every $0<\gamma<180^\circ$, the graph $\Theta_4$ is not angle-monotone with width $\gamma$.

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