Let G be a spanning subgraph of Ks,s and let H be the complement of G relative to Ks,s; that is, Ks,s = GtJ)H is a factorization of Ks,s' The graph G is ,,(-critical relative to Ks,s if ,(G) = , and "(( G + e) = , 1 for all e E E(H), where ,(G) denotes the domination number of G. We investigate ,,(-critical graphs for small values of "(. The 2-critical graphs and 3-critical graphs are character...

A critical element in the any repair or upgrading project is the need to assess the condition of the structure. This assessment is needed to establish the existing condition of the structure and to plan for its repair. Secondly, means to assess in real time the progress of the repair are required especially when techniques such as grouting are being used. Finally, an assessment of the modified ...

A graph G is said to be k-γt-critical if the total domination number γt(G) = k and γt(G + uv) < k for every uv / ∈ E(G). A k-γc-critical graph G is a graph with the connected domination number γc(G) = k and γc(G + uv) < k for every uv / ∈ E(G). Further, a k-tvc graph is a graph with γt(G) = k and γt(G− v) < k for all v ∈ V (G), where v is not a support vertex (i.e. all neighbors of v have degre...

Youngs, D.A., Gallai’s problem on Dirac’s construction, Discrete Mathematics 101 (1992) 343-350. It is thought that T. Gallai posed the following problem concerning a construction due to G.A. Dirac: suppose that a graph K consists of disjoint subgraphs G and Hand a set of edges joining them. If each of G, H, and K are colour critical graphs, under what circumstances is it then true that the joi...

Corporate culture, as a management theory and method, emerged as an important component of organizational success in the late twentieth century. Korea’s unique style of corporate culture is a critical element to its economic success. Both China and Korea still have a strong influence of Confucianism on everyday life; Confucianism is the most dominating component in both countries’ cultures. The...

Let G be a spanning subgraph of K(s, s) and let H be the complement of G relative to K(s, s); that is, K(s, s) = G(f)H is a factorization of K(s, s). The graph G is ,),-relative-critical if ')'(G) = ')' and ')'(G + e) = "( 1 for all e E E(H), where "((G) denotes the domination number of G. The 2-relative-critical graphs and 3-relative-critical graphs are characterized in [7]. In [7], it is show...

We show that the diameter of a total domination vertex-critical graph is at most 5(γt −1)/3, and that the diameter of an independent domination vertex-critical graph is at most 2(i− 1). For all values of γt ≡ 2 (mod 3) there exists a total domination vertex-critical graph with the maximum possible diameter. For all values of i ≥ 2 there exists an independent domination vertex-critical graph wit...

The recognition of conceptual knowledge as a critical element in effective content-area instruction has been advanced by the development of concept mapping applications to instruction. This paper broadens awareness of how concept mapping can serve as a basis for linking instructional strategies that when combined with concept mapping form an instructional system teachers are able to use to impr...

2 as a critical element of the United States patent system. The requirement mandates that, at the time of invention, an invention must not have been obvious to a person having ordinary skill in the art. In effect, this screens routine improvements out of the patent system and rewards inventions perceived to be scientific breakthroughs. Recent Supreme Court cases developing non-obviousness highl...

Borodin and Kostochka conjectured that every graph G with maximum degree ∆ ≥ 9 satisfies χ ≤ max {ω,∆− 1}. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main tool is the classification of graph joins A ∗B with |A| ≥ 2, |B| ≥ 2 which are f -choosable, where f(v) := d(v)− 1 for each vertex v. Since such a join cannot be an induced subgraph of a...