arXiv subjects
Publications and source records attributed to Ajit Arvind Diwan.
In this paper we show that it can be decided in polynomial time whether or not the visibility graph of a given point set is 4-colourable, and such a 4-colouring, if it exists, can also be constructed in polynomial time. We show that the problem of deciding whether the visibility graph of a point set is 5-colourable, is NP-complete. We give an example of a point visibility graph that has chromatic number 6 while its clique number is only 4.
In this paper, we characterize planar point sets that can be partitioned into disjoint polygons of arbitrarily specified sizes. We provide an algorithm to construct such a partition, if it exists, in polynomial time. We show that this problem is equivalent to finding a specified $2$-factor in the visibility graph of the point set. The characterization for the case where all cycles have length $3$ also translates to finding a $K_3$-factor of the visibility graph of the point set. We show that the generalized problem of finding a $K_k$-factor of the visibility graph of a given point set for $k \geq 5$ is NP-hard.
In this paper, we consider the problem of determining in polynomial time whether a given planar point set $P$ of $n$ points admits 4-connected triangulation. We propose a necessary and sufficient condition for recognizing $P$, and present an $O(n^3)$ algorithm of constructing a 4-connected triangulation of $P$. Thus, our algorithm solves a longstanding open problem in computational geometry and geometric graph theory. We also provide a simple method for constructing a noncomplex triangulation of $P$ which requires $O(n^2)$ steps. This method provides a new insight to the structure of 4-connected triangulation of point sets.
In this paper, we establish two necessary conditions for a joint triangulation of two sets of $n$ points in the plane and conjecture that they are sufficient. We show that these necessary conditions can be tested in $O(n^3)$ time. For the problem of a joint triangulation of two simple polygons of $n$ vertices, we propose an $O(n^3)$ time algorithm for constructing a joint triangulation using dynamic programming.