Algorithmic Study on Position and Movement Method of Badminton Doubles

The algorithms used by schedulers depend on the complexity of the schedule and constraints for each problem. The position and movement of badminton players in badminton doubles competition is one of the key factors to improve the athletes’ transition efficiency of offense and defense and the rate of winning matches and to save energy consumption. From the perspective of basic theory, the author conducts research on the position and movement of badminton doubles. Based on the numerical analysis method, the optimal model of standing position and direction composed of 7 nonlinear equations is established. In addition, the final of 10 matches of the super series of the world badminton federation in 2019 was selected as the sample of speed parameters. With the help of MATLAB mathematical analysis software, the numerical model established by the least square method was adopted to optimize the specific standing position and walking model. Ultimately, the optimal solution has been obtained, which can be represented on a plane graph. The optimal position of the attack station should be the blocking area (saddle-shaped area) and the hanging area (circular arc area in the middle). The optimal defensive positioning should be left defensive positioning area (left front triangle area) and right defensive positioning area (right front triangle area), which is consistent with our current experience and research results. The research results use mathematical tools to calculate the accurate optimal position in doubles matches, which has guiding significance to the choice of athletes’ position and walking position in actual combat and can also be used as a reference for training, providing a certain theoretical basis for the standing and walking of badminton doubles confrontation. The data collection and operation methods in this study can provide better calculation materials for artificial intelligence optimization and fuzzy operation of motion displacement, which is of great significance in the field of motion, simulation, and the call of parametric functions.

A New Algorithm for Embedding Plane Graphs at Fixed Vertex Locations

We show that a plane graph can be embedded with its vertices at arbitrarily assigned locations in the plane and at most $6n-1$ bends per edge. This improves and simplifies a classic result by Pach and Wenger. The proof extends to orthogonal drawings.

Total Face Irregularity Strength of Grid and Wheel Graph under K-Labeling of Type (1, 1, 0)

In this study, we used grids and wheel graphs G = V , E , F , which are simple, finite, plane, and undirected graphs with V as the vertex set, E as the edge set, and F as the face set. The article addresses the problem to find the face irregularity strength of some families of generalized plane graphs under k -labeling of type α , β , γ . In this labeling, a graph is assigning positive integers to graph vertices, graph edges, or graph faces. A minimum integer k for which a total label of all verteices and edges of a plane graph has distinct face weights is called k -labeling of a graph. The integer k is named as total face irregularity strength of the graph and denoted as tfs G . We also discussed a special case of total face irregularity strength of plane graphs under k -labeling of type (1, 1, 0). The results will be verified by using figures and examples.

On the Circumference of 3-Connected Cubic Triangle-Free Plane Graphs

The circumference of a graph G is the length of a longest cycle in G , denoted by cir G . For any even number n , let c n = min { cir G | G is a 3-connected cubic triangle-free plane graph with n vertices}. In this paper, we show that an upper bound of c n is n + 1 − 3 ⌊ n / 136 ⌋ for n ≥ 136 .

Eulerian and Even-Face Graph Partial Duals

Eulerian and bipartite graph is a dual symmetric concept in Graph theory. It is well-known that a plane graph is Eulerian if and only if its geometric dual is bipartite. In this paper, we generalize the well-known result to embedded graphs and partial duals of cellularly embedded graphs, and characterize Eulerian and even-face graph partial duals of a cellularly embedded graph by means of half-edge orientations of its medial graph.

Facial Homogeneous Colouring of Graphs

A proper colouring of a plane graph G is called facially homogeneous if it uses the same number of colours for every face of G. We study various sufficient conditions of facial homogeneous colourability of plane graphs, its relation to other facial colourings, and the extension of this concept for embedded graphs in general.

A proof of four color conjecture (4CC)

foundout a critical corresponding relationship between a δwheelof any maximal simple plane graph of n vertices(nMSPG)and its corresponding δcycleof a corresponding n-1MSPGof the nMSPG, and relied on theinductive way, while with the help of KempeChain (KC), this article gives a brand-newlogical proof for the famous and morethan 100 years old fourcolor conjecture (4CC).

Graph Planarity by Replacing Cliques with Paths

This paper introduces and studies the following beyond-planarity problem, which we call h-Clique2Path Planarity. Let G be a simple topological graph whose vertices are partitioned into subsets of size at most h, each inducing a clique. h-Clique2Path Planarity asks whether it is possible to obtain a planar subgraph of G by removing edges from each clique so that the subgraph induced by each subset is a path. We investigate the complexity of this problem in relation to k-planarity. In particular, we prove that h-Clique2Path Planarity is NP-complete even when h=4 and G is a simple 3-plane graph, while it can be solved in linear time when G is a simple 1-plane graph, for any value of h. Our results contribute to the growing fields of hybrid planarity and of graph drawing beyond planarity.