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Automatic hexahedral mesh generation plays a fundamental role in CAD/CAE fields, especially for IGA (isogeometric analysis). Recently, an automatic hex-mesh generation method has been proposed, which is based on the equivalence among three key concepts: colorable quadrilateral meshes, finite measured foliations and Strebel differentials. This work focuses on the computational aspect of Strebel differentials on high genus surfaces, which is based on graph-valued harmonic mapping. The algorithmic pipeline is as follows: first, the user inputs an admissible curve system, which induces a cylindric-decomposition graph; then the user specifies the lengths of the edges of the graph; third, the algorithm finds the unique harmonic map from the surface to the metric graph by a non-linear heat flow; finally, the harmonic map induces a Strebel differential, which can be further utilized to generate the hex-mesh. The method has solid theoretic foundation, which guarantees the existence and the uniqueness of the graph-valued harmonic map. The algorithm is capable of handling surfaces with complicated topologies, and producing all possible Strebel differentials on the surface. The user has full control of the combinatorial type and the geometry of the Strebel differential. The computational pipeline is automatic. The experimental results show the efficiency and efficacy of the algorithm, and demonstrate the great potential for automating the structured hexahedral mesh generation.

In his article “Defend the Roman Empire!” (1999), Ian Stewart discussed a strategy of Emperor Constantine for defending the Roman Empire. Motivated by this article, Cockayne et al. (2004) introduced the notion of Roman domination in graphs. Let 𝐺 = (𝑉, 𝐸) be a graph. A Roman dominating function of 𝐺 is a function 𝑓 ∶ 𝑉 → {0, 1, 2} such that every vertex 𝑣 for which 𝑓(𝑣) = 0 has a neighbor 𝑢 with 𝑓(𝑢) = 2. The weight of a Roman dominating function 𝑓 is 𝑤(𝑓) =∑_𝑣∈𝑉 𝑓(𝑣). The Roman domination number of a graph 𝐺, denoted by 𝛾_𝑅(𝐺), is the minimum weight of all possible Roman dominating functions. This paper introduces a quantity 𝑅(𝑥𝑦) for each pair of non-adjacent vertices {𝑥, 𝑦 }in 𝐺, called the Roman dominating index of{ 𝑥, 𝑦} , which is defined by &𝑅(𝑥𝑦) = 𝛾_𝑅(𝐺) − 𝛾_𝑅(𝐺 + 𝑥𝑦)&. We prove that 0 ≤ 𝑅(𝑥𝑦) ≤ 1 and give a necessary and sufficient condition on {𝑥, 𝑦} for which 𝑅(𝑥𝑦)= 1. This paper also introduces the Roman-critical graph. We call 𝐺 = (𝑉, 𝐸) Roman-critical if 𝛾_𝑅(𝐺 − 𝑒) > 𝛾_𝑅(𝐺) , ∀𝑒 ∈ 𝐸 . It is proved that a Roman-critical graph can only be a star graph whose order is not equal to 2, or the union of such graphs. In addition, this paper shows that for each connected graph 𝐺 of order& 𝑛 ≥ 3,2 ≤ 𝛾_𝑅(𝐺) ≤frac{4𝑛}{5}&. A family of graphs for which the respective equality holds is also provided. Finally, this paper finds the lower bound of the Roman domination number for 3-regular graphs and the exact value of the Roman domination number for &𝐶_frac{𝑛}{2}*𝑃2& and 3-regular circulant graphs.

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