Counting Plane Graphs: Flippability and its Applications

Counting Plane Graphs: Cross-Graph Charging Schemes

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of triangulations that are embedded over a fixed set of points in the plane. We show how this method can be generalized to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound of $O^*(187.53^N)$ (where the $O^*()$ notation hides polynomial factors) for the maximum number of crossing-free straight-edge graphs th…

Counting Plane Graphs: Perfect Matchings, Spanning Cycles, and Kasteleyn's Technique

We derive improved upper bounds on the number of crossing-free straight-edge spanning cycles (also known as Hamiltonian tours and simple polygonizations) that can be embedded over any specific set of $N$ points in the plane. More specifically, we bound the ratio between the number of spanning cycles (or perfect matchings) that can be embedded over a point set and the number of triangulations that can be embedded over it. The respective bounds are $O(1.8181^N)$ for cycles and $O(1.1067^N)$ for matchings. These imply a new upper bound of $O(54.543^N)$ on the number of crossing-free straight-edge…

Counting embeddings of rooted trees into families of rooted trees

The number of embeddings of a partially ordered set $S$ in a partially ordered set $T$ is the number of subposets of $T$ isomorphic to $S$. If both, $S$ and $T$, have only one unique maximal element, we define good embeddings as those in which the maximal elements of $S$ and $T$ overlap. We investigate the number of good and all embeddings of a rooted poset $S$ in the family of all binary trees on $n$ elements considering two cases: plane (when the order of descendants matters) and non-plane. Furthermore, we study the number of embeddings of a rooted poset $S$ in the family of all planted plan…

Counting Carambolas

We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of $n$ points in the plane. Configurations of interest include \emph{convex polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also consider related problems for \emph{directed} planar straight-line graphs.

Counting Hexagonal Patches and Independent Sets in Circle Graphs

A hexagonal patch is a plane graph in which inner faces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the study of benzenoid hydrocarbons and fullerenes in computational chemistry. We give the first polynomial time algorithm for this problem. We show that it can be reduced to counting maximum independent sets in circle graphs, and give a simple and fast algorithm fo…

计数巧合的计算复杂度

Computational complexity of counting coincidences

该研究聚焦组合对象计数的巧合判定问题，证明两类多米诺铺砌的巧合问题不在多项式谱系内（除非谱系坍缩），并拓展至多类组合对象的相关研究与猜想。