The Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.

A general assumption made in evaluation of flow network reliability is perfectly reliable nodes. Under such assumption, network reliability is evaluated through subset cut or composite path set approaches. This paper presents a simple approach to account for node failures in reliability evaluation of flow networks using subset cut approach. The proposed approach starts with generating combinati...

Reduced-order models (ROMs) for turbulent combustion rely on identifying a small number of parameters that can effectively describe the complexity reacting flows. With advent data-driven approaches, ROMs be trained datasets representing thermo-chemical state-space in simple systems. For low-Mach flows, full state vector serves as training dataset is typically composed temperature and chemical c...

We generalize a well-known algorithm for the generation of all subsets of a set in lexicographic order with respect to the sets as lists of elements (subset-lex order). We obtain algorithms for various combinatorial objects such as the subsets of a multiset, compositions and partitions represented as lists of parts, and for certain restricted growth strings. The algorithms are often loopless an...

Given a family F of k sets with cardinalities s1, s2, . . . , sk and N = ∑ k i=1 si, we show that the size of the partial order graph induced by the subset relation (called the subset graph) is O( ∑ si≤B 2i + N/ logN · ∑ si>B log (2si/B)), where B = log (N/ log N). This implies a simpler proof to the O(N/ log N) bound concluded in [2]. We also give an algorithm that computes the subset graph fo...

Let k be a positive integer. A vertex subset D of a graph G = (V,E) is a perfect k-dominating set of G if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γkp(G). In this paper, we generalize perfect domination to perfect k-domination, where many bounds of γkp(G) are obtained. We ...

This note is to report the existence of a simple perfect dissection of a square into 21 unequal squares.

A finite group G is said to have Perfect Order Subsets if for every d, the number of elements of G of order d (if there are any) divides |G|. This notion was introduced in the paper [1] by C. Finch and L. Jones. In the case of finite Abelian groups, the authors reduced the problem of which groups have this property to the case of groups of the form G = ∏k i=1(Z/piZ) ai , where pi are primes and...

We report the existence of two simple perfect square-cylinders of order 20 shown in Figs. 1 and 2. Dissected cylinders are associated with graphs which we call cyl-nets. A cyl-net is obtained from a dissected cylinder in the following way. The two rims of the cylinder are associated with two vertices which are called “source” and “sink,” respectively. Each horizontal line is associated with a v...