An Algorithmic Solution of a Birkhoff Type Problem
Fundamenta Informaticae
Star Coloring and Acyclic Coloring of Locally Planar Graphs
SIAM Journal on Discrete Mathematics
Experiences in Computing Mesh Root Systems for Dynkin Diagrams Using Maple and C++
SYNASC '11 Proceedings of the 2011 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
Mesh Algorithms for Solving Principal Diophantine Equations, Sand-glass Tubes and Tori of Roots
Fundamenta Informaticae
Rainbow Induced Subgraphs in Proper Vertex Colorings
Fundamenta Informaticae
Tree Matrices and a Matrix Reduction Algorithm of Belitskii
Fundamenta Informaticae
Inflation Algorithms for Positive and Principal Edge-bipartite Graphs and Unit Quadratic Forms
Fundamenta Informaticae
On Computing Mesh Root Systems and the Isotropy Group for Simply-laced Dynkin Diagrams
SYNASC '12 Proceedings of the 2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
On the Computational Complexity of Bongartz's Algorithm
Fundamenta Informaticae
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By applying symbolic and numerical computation and the spectral Coxeter analysis technique of matrix morsifications introduced in our previous paper [Fund. Inform. 1242013], we present a complete algorithmic classification of the rational morsifications and their mesh geometries of root orbits for the Dynkin diagram $\mathbb{D}_4$. The structure of the isotropy group $Gl4, \mathbb{Z}_{\mathbb{D}_4}$ of $\mathbb{D}_4$ is also studied. As a byproduct of our technique we show that, given a connected loop-free positive edge-bipartite graph Δ, with n ≥ 4 vertices in the sense of our paper [SIAM J. Discrete Math. 272013] and the positive definite Gram unit form $q_\Delta : \mathbb{Z}^n \rightarrow \mathbb{Z}$, any positive integer d ≥ 1 can be presented as d = qΔv, with $v \in \mathbb{Z}^n$. In case n = 3, a positive integer d ≥ 1 can be presented as d = qΔv, with $v \in \mathbb{Z}^n$, if and only if d is not of the form 4a16 · b + 14, where a and b are non-negative integers.