Tangle and Ultrafilter: Game Theoretical Interpretation

Abstract

The investigation of width parameters in both graph and algebraic contexts has attracted considerable interest. Among these parameters, the linear branch width has emerged as a crucial measure. In this concise paper, we explore the concept of linear decomposition, specifically focusing on the single filter in a connectivity system. Additionally, we examine the relevance of matroids, antimatroids, and greedoids in the context of connectivity systems. Our primary objective in this study is to shed light on the impediments to linear decomposition from multiple perspectives.

Ultrafilter in Digraph: Directed Tangle and Directed Ultrafilter

“…researchers, as indicated by references [1,20,21,22,[23][24][25][26][27][28][29][30][31][32][33][34][35]36,37,38]. This widespread engagement underscores the perceived value of ultrafilter studies.…”

Filter for Submodular Partition Function: Connection to Loose Tangle

“…We will consider about single ideal [19] and linear tangle [20] using Submodular Partition Function. Also we will discuss about ultrafilter [7,11], tangle [10,32] using Submodular Partition Function.…”

“…The investigation of graph width parameters finds extensive applications across diverse fields, such as matroid theory, lattice theory, theoretical computer science, game theory, network theory, artificial intelligence, graph theory, and discrete mathematics, as evidenced by numerous studies (for example, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]22,[28][29][30][31][32][33]). These graph width parameters are frequently explored in conjunction with obstruction, contributing to a robust body of research.…”

Alternative Proof of Linear Tangle and Linear Obstacle: An Equivalence Result

“…A strategy in which the player always emerges victorious is termed a "winning strategy". This winning strategy is characterized by various width parameters and their dual concepts commonly employed in graph theory (e.g., [3,19,20,21]). For instance, the concept of (k, m)-obstacle on connectivity systems is proposed in reference [3].…”