Equivalence between Linear Tangle and Maximal Single Ideal

Abstract

The concept of linear tangle was introduced as an obstruction to mixed searching number. The concept of single ideal has been introduced as an obstruction to linear-width. Moreover, it was already known that mixed search number is equivalent to linear-width. Hence, by combining those results, we obtain a proof of the equivalence between linear tangle and single ideal. This short report gives an alternative proof of the equivalence.

Novel Idea on Edge-Ultrafilter and Edge-Tangle

“…In this paper, we introduced a novel definition termed Edge-Ultrafilters on graphs and demonstrated their equivalence to Edge-tangles. Furthermore, we intend to investigate the relationship between linear tangles [14,15,16] and edge-tangles in future research.…”

Tangle and Ultrafilter: Game Theoretical Interpretation

“…Theorem 9 [31]: Let X be a finite set and f be a symmetric submodular function. Under the assumption that f({e})≤k for every e∈X, F is a linear tangle of order k+1 on (X,f) iff F is a linear obstacle of order k+1 on (X,f).…”

“…In recent years, researchers have shown very significant interest in studying width parameters in both graph theory and algebraic contexts [1,2,3,4,5,6,7,8,9,[12][13][14]17,21,22,[28][29][30][31][32][33][34][48][49][50]. Width parameters refer to parameters obtained from tree-like structures known as decompositions on graphs.…”

“…One important width parameter is the linear branch width, which has been shown to be a crucial measure in characterizing the complexity of various mathematical objects, including graphs and matroids (cf. [31][32][33]51]). "Linear tangle" is a concept proposed by Bienstock et al in literature [10] as a strategy for cops and robbers.…”

Filter for Submodular Partition Function: Connection to Loose Tangle

“…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.…”