In a graph, a set \(D\) is an \(n\)-dominating set if for every vertex \(x\), not in \(D\), \(x\) is adjacent to at least \(n\) vertices of \(D\). The \(n\)-domination number, \(\gamma_n(G)\), is the order of a smallest \(n\)-dominating set. When this concept was first introduced by Fink and Jacobson, they asked whether there existed a function \(f(n)\), such that if \(G\) is any graph with minimum degree at least \(n\), then \(\gamma_n(G) < \gamma_{f(n)}(G)\). In this paper we show that \(\gamma_2(G) < \gamma_5(G)\) for all graphs with minimum degree at least \(2\). Further, this result is best possible in the sense that there exist infinitely many graphs \(G\) with minimum degree at least \(2\) having \(\gamma_2(G) = \gamma_4(G)\).

Inclusive connectivity parameters for a given vertex in a graph \(G\) are measures of how close that vertex is to being a cutvertex. Thus they provide a local measure of graph vulnerability. In this paper we provide bounds on the inclusive connectivity parameters in \(K_2 \times G\) and inductively extend the results to a certain generalized hypercube.

In this paper, the maximum graphical structure is obtained when the number of vertices p of a connected graph G and tenacity \(T(G) = T\) are given. Finally, the method of constructing the sort of graphs is also presented.

Let \(G\) be a bipartite graph with bipartite sets \(V_1\) and \(V_2\). If \(f\) is a bijective function from the vertices and edges of \(G\) into the first \(p+q\) positive integers, where \(p\) and \(q\) denote the order and size of \(G\), respectively, meeting the properties that \(f\) is a super edge magic labeling and if the cardinal of \(V_i\) is \(p_i\) for \(i=1,2\), then the image of the set \(V_1\) is the set of the first \(p_i\) positive integers and the image of the set \(V_2\) is the set of integers from \(p_1 + 1\) up to \(p\). If a bipartite graph \(G\) admits an special super edge magic labeling, we say that \(G\) is special super edge magic. Some properties of special super edge magic graphs are presented. However, this work is mainly devoted to the study of the relations existing between super edge magic and special super edge magic labelings.

In this note, we present necessary conditions for decomposing \(\lambda K_n\) into copies of \(K_{2,5}\), and show that these conditions are sufficient except for \(\lambda = 5\) and \(n = 8\), and possibly for the following cases: \(\lambda = 1\) and \(n = 40\); and \(\lambda = 3\) and \(n = 16\) or \(20\).

We obtain \(135\) nonisomorphic nearly Kirkman triple systems of order \(18\) (the smallest order for which such a system exists), including all \(119\) systems of a well-defined subclass.

In overloaded task systems, it is by definition not possible to complete all tasks by their deadlines. However, it may still be desirable to maximize the number of in-time task completions. The performance of on-line schedulers with respect to this metric is investigated here. It is shown that in general, an on-line algorithm may perform arbitrarily poorly as compared to clairvoyant (off-line) schedulers. This result holds for general task workloads where there are no constraints on task characteristics. For a variety of constrained workloads that are representative of many practical applications, however, on-line schedulers that do provide a guaranteed level of performance are presented.

We present a new algorithm for computer searches for orthogonal designs. Then we use this algorithm to find new sets of sequences with entries from \(\{0,\pm a, \pm b, \pm c,\pm d\}\) on the commuting variables \(a, b, c, d\) with zero autocorrelation function.

Consider the hit polynomial of the path \(P_{2n}\) embedded in the complete graph \(K_{2n}\). We give a combinatorial interpretation of the \(n\)-th Bessel polynomial in terms of a modification of this hit polynomial, called the ordered hit polynomial. Also, the first derivative of the \(n\)-th Bessel polynomial is shown to be the ordered hit polynomial of \(P_{2n-1}\) embedded in \(K_{2n}\).

In a packer-spoiler game on a graph, two players jointly construct a maximal partial \(F\)-packing of the graph according to some rules, where \(F\) is some given graph. The packer wins if all the edges are used up and the spoiler wins otherwise. The question of which graphs are wins for which player generalizes the questions of which graphs are \(F\)-packable and which are randomly \(F\)-packable. While in general such games are NP-hard to solve, we provide partial results for \(F = P_3\) and solutions for \(F = 2K_2\).