اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدCharacterizing Graphs of Zonohedra
329
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
A classic theorem by Steinitz states that a graph G is realizable by a convex polyhedron if and only if G is 3-connected planar. Zonohedra are an important subclass of convex polyhedra having the property that the faces of a zonohedron are parallelograms and are in parallel pairs. In this paper we give characterization of graphs of zonohedra. We also give a linear time algorithm to recognize such a graph. In our quest for finding the algorithm, we prove that in a zonohedron P both the number of zones and the number of faces in each zone is O(square root{n}), where n is the number of vertices of P.
قيم البحث
اقرأ أيضاً
We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of $n$ unit disks in the plane there exists a line $ell$ such that $ell$ intersects at most $O(sqrt{(m+n)log{n}})$ disks and each of the hal
fplanes determined by $ell$ contains at most $2n/3$ unit disks from the set, where $m$ is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting $O(sqrt{m+n})$ disks exists, but each halfplane may contain up to $4n/5$ disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in $n$ exists when we look at disks of arbitrary radii, even when $m=0$. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size $O(sqrt{m})$).
In the Art Gallery Problem we are given a polygon $Psubset [0,L]^2$ on $n$ vertices and a number $k$. We want to find a guard set $G$ of size $k$, such that each point in $P$ is seen by a guard in $G$. Formally, a guard $g$ sees a point $p in P$ if t
he line segment $pg$ is fully contained inside the polygon $P$. The history and practical findings indicate that irrational coordinates are a very rare phenomenon. We give a theoretical explanation. Next to worst case analysis, Smoothed Analysis gained popularity to explain the practical performance of algorithms, even if they perform badly in the worst case. The idea is to study the expected performance on small perturbations of the worst input. The performance is measured in terms of the magnitude $delta$ of the perturbation and the input size. We consider four different models of perturbation. We show that the expected number of bits to describe optimal guard positions per guard is logarithmic in the input and the magnitude of the perturbation. This shows from a theoretical perspective that rational guards with small bit-complexity are typical. Note that describing the guard position is the bottleneck to show NP-membership. The significance of our results is that algebraic methods are not needed to solve the Art Gallery Problem in typical instances. This is the first time an $existsmathbb{R}$-complete problem was analyzed by Smoothed Analysis.
Barnette identified two interesting classes of cubic polyhedral graphs for which he conjectured the existence of a Hamiltonian cycle. Goodey proved the conjecture for the intersection of the two classes. We examine these classes from the point of vie
w of distance-two colorings. A distance-two $r$-coloring of a graph $G$ is an assignment of $r$ colors to the vertices of $G$ so that any two vertices at distance at most two have different colors. Note that a cubic graph needs at least four colors. The distance-two four-coloring problem for cubic planar graphs is known to be NP-complete. We claim the problem remains NP-complete for tri-connected bipartite cubic planar graphs, which we call type-one Barnette graphs, since they are the first class identified by Barnette. By contrast, we claim the problem is polynomial for cubic plane graphs with face sizes $3, 4, 5,$ or $6$, which we call type-two Barnette graphs, because of their relation to Barnettes second conjecture. We call Goodey graphs those type-two Barnette graphs all of whose faces have size $4$ or $6$. We fully describe all Goodey graphs that admit a distance-two four-coloring, and characterize the remaining type-two Barnette graphs that admit a distance-two four-coloring according to their face size. For quartic plane graphs, the analogue of type-two Barnette graphs are graphs with face sizes $3$ or $4$. For this class, the distance-two four-coloring problem is also polynomial; in fact, we can again fully describe all colorable instances -- there are exactly two such graphs.
Given a convex polyhedron $P$ of $n$ vertices inside a sphere $Q$, we give an $O(n^3)$-time algorithm that cuts $P$ out of $Q$ by using guillotine cuts and has cutting cost $O((log n)^2)$ times the optimal.
This is the arXiv index for the electronic proceedings of GD 2019, which contains the peer-reviewed and revised accepted papers with an optional appendix. Proceedings (without appendices) are also to be published by Springer in the Lecture Notes in Computer Science series.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
