Computing the forcing and anti-forcing numbers of perfect matchings for graphs by integer linear programmings

The forcing polynomial and anti-forcing polynomial are two important enumerative polynomials associated with all perfect matchings of a graph. In a graph with large order, the exhaustive enumeration which is used to compute forcing number of a given perfect matching is too time-consuming to compute anti-forcing number. In this paper, we come up with an efficient method — integer linear programming, to compute forcing number and anti-forcing number of a given perfect matching. As applications, we obtain the di-forcing polynomials C60 , C70 and C72 , and as a consequence, the forcing and anti-forcing polynomials of them are obtained.

Order-Independent Algorithm for the Asymptotic Stability of Complex Polynomials

The Extended Routh Array (ERA) settles the asymptotic stability of complex polynomials. The ERA is a natural extension of the Routh Array which applies only to real polynomials. Although the ERA is a nice theoretical algorithm for stability testing, it has its limitations. Unfortunately, as the order of the polynomial increases, the size of calculations increases dramatically as will be shown below. In the current work, we offer an alternative algorithm which is basically equivalent to the ERA, but has the extra advantage of being simpler, more efficient, and easy to apply even to large order polynomials. In all the steps required in the construction of the new algorithm, only one single and simple algebraic operation is needed, which makes it a polynomial order-independent algorithm.

The combinatorial analysis of n-gram dictionaries, coverage and information entropy based on the web corpus of English

We research n-gram dictionaries and estimate its coverage and entropy based on the web corpus of English. We consider a method for estimating the coverage of empirically generated dictionaries and an approach to address the disadvantage of low coverage. Based on the ideas of Kolmogorov’s combinatorial approach, we estimate the n-gram entropy of the English language and use mathematical extrapolation to approximate the marginal entropy. In addition, we approximate the number of all possible legal n-grams in the English language for large order of n-grams.

Reconfiguring Dominating Sets in Minor-Closed Graph Classes

For a graph G, two dominating sets D and $$D'$$ D ′ in G, and a non-negative integer k, the set D is said to k-transform to $$D'$$ D ′ if there is a sequence $$D_0,\ldots ,D_\ell $$ D 0 , … , D ℓ of dominating sets in G such that $$D=D_0$$ D = D 0 , $$D'=D_\ell $$ D ′ = D ℓ , $$|D_i|\le k$$ | D i | ≤ k for every $$i\in \{ 0,1,\ldots ,\ell \}$$ i ∈ { 0 , 1 , … , ℓ } , and $$D_i$$ D i arises from $$D_{i-1}$$ D i - 1 by adding or removing one vertex for every $$i\in \{ 1,\ldots ,\ell \}$$ i ∈ { 1 , … , ℓ } . We prove that there is some positive constant c and there are toroidal graphs G of arbitrarily large order n, and two minimum dominating sets D and $$D'$$ D ′ in G such that Dk-transforms to $$D'$$ D ′ only if $$k\ge \max \{ |D|,|D'|\}+c\sqrt{n}$$ k ≥ max { | D | , | D ′ | } + c n . Conversely, for every hereditary class $$\mathcal{G}$$ G that has balanced separators of order $$n\mapsto n^\alpha $$ n ↦ n α for some $$\alpha <1$$ α < 1 , we prove that there is some positive constant C such that, if G is a graph in $$\mathcal{G}$$ G of order n, and D and $$D'$$ D ′ are two dominating sets in G, then Dk-transforms to $$D'$$ D ′ for $$k=\max \{ |D|,|D'|\}+\lfloor Cn^\alpha \rfloor $$ k = max { | D | , | D ′ | } + ⌊ C n α ⌋ .

Procurement modes for emergency supplies in the presence of disaster and commercial demands

This paper studies alternative procurement modes for emergency supplies in the presence of a public sector and a private supplier. A key feature of such a supply chain is that the private supplier must consider commercial demand in addition to disaster demand. Three procurement modes are analysed: an option mode (OM), an order-before-disaster mode (OBDM) and a procurement-after-disaster mode (PADM). We provide the optimal decisions associated with these three modes. Theoretical results in the OM show that a large order is not always better for the private supplier. From theoretical and numerical comparative analyses, no mode is absolutely superior. For the public sector, there are two thresholds. When the disaster probability is less than the low threshold, the revenues of all procurement modes are the same; when the disaster probability is larger than the high threshold, the OBDM has the highest revenue; otherwise, the OM has the highest revenue. However, for the private supplier, the PADM always has the highest revenue.