Suppose \(S\) is a defining set of a symmetric \(2\)-( \(v, k, \lambda\) ) design \(D\), where \(\lambda = 1\) or \(2\); that is, \(D\) is a projective plane or a biplane.In this paper, conditions under which the residual of \(S\) is a defining set of the residual of \(D\) are investigated.As a consequence, inequalities relating the sizes of smallest defining sets of \(D\) and of the residual of \(D\) are obtained.The exact sizes of smallest defining sets of \({PG}(2, 5)\), \({AG}(2, 5)\), and the three non-isomorphic \(2\)-( \(10, 4, 2\) ) designs are determined.

Exact designs with \(n\) observations and \(k\) two-level factors in the presence of autocorrelated errors are considered. The problem of finding \(D\)- and \(A\)- optimal designs is discussed. An algorithm for constructing such designs, using exhaustive search for different values of \(n\) and \(k\), is developed. The application of this algorithm showed that, in the case of positive autocorrelation, the maximum possible number of interchanges of the factor levels provides almost optimal designs.On the contrary, in the case of negative autocorrelation, the minimum such number provides almost optimal designs. A list of the exact \(D\)- and \(A\)-optimal designs is given.

A tree \(T\) consisting of a line with edges \(\{(1, 2), (2, 3), \ldots, (n-1, n)\}\) and with edges \(\{(1, a_1), (1, a_2), \ldots, (1, a_k)\}\) (a star) attached on the left, is called a broom.
The edges of the tree \(T\) are called \(T\)-transpositions. We give an algorithm to factor any permutation \(\sigma\) of \(\{a_1, a_2, \ldots, a_k, 1, 2, \ldots ,n\}\) as a product of \(T\)-transpositions, and prove that the factorization produced by the algorithm has minimal length.

Median graphs are surveyed from the point of view of their characterizations, their role in location theory, and their connections with median structures. The median structures we present include ternary algebras, betweenness, interval structures, semilattices, hypergraphs, join geometries, and conflict models. In addition, some new characterizations of median graphs as meshed graphs are presented and a new characterization in terms of location theory is given.

Up to isomorphisms, there are exactly 22 \(1\)-rotational resolved \((52,4,1)\)-BIBD’s.

Let \(\mathcal{F}\) be a family of objects and \(\varphi\) an integer-valued function defined on \(\mathcal{F}\).If for any \(A, B \in \mathcal{F}\) and integer \(k\) between \(\varphi(A)\) and \(\varphi(B)\), there exists \(C \in \mathcal{F}\) such that \(\varphi(C) = k\), then \(\varphi\) is said to interpolate over \(\mathcal{F}\).In this paper, we first discuss some basic ideas used in proving interpolation theorems for graphs.By using this, we then prove that a number of conditional invariants interpolate over some families of subgraphs of a given connected graph.

Scheduling graphs are used by algorithms such as PERT/CPM in order to determine an optimal schedule for a given project. It is well-known that dummy tasks (requiring zero processing time) must sometimes be incorporated into a scheduling graph.

The main tool in this paper is a new algorithm, RESOLVE, which creates a scheduling graph, typically with fewer dummy tasks than are produced by Richards’ algorithm (1967). A theoretical framework for scheduling graphs is systematically developed through several theorems, culminating in a demonstration of the validity of RESOLVE. A worked example illustrating the application of RESOLVE concludes the paper.

Let \(\mathcal{A} = \{A_1, \ldots, A_l\}\) be a partition of \([n]\) and \(\mathcal{F} = \{S_1, \ldots, S_m\}\) be an intersecting family of distinct nonempty subsets of \([n]\) such that \(\mathcal{A}\) and \(\mathcal{F}\) are pairwise intersecting families.Then \(|\mathcal{F}| \leq \frac{1}{2} \prod_{i=1}^{l} \left( 2^{|A_i|} – 2 \right) + \sum_{S\subsetneqq[l]} \left(\prod_{i\in S}\left( 2^{|A_i|} – 2 \right)\right).\)From this result and some properties of intersection graphs on multifamilies, we determine the intersection numbers of \(3\), \(4\), and \(5\)-regular graphs and some special graphs.

The concept of tenacity of a graph \(G\) was introduced in References [5,6] as a useful measure of the “vulnerability” of \(G\). In assessing the “vulnerability” of a graph, one determines the extent to which the graph retains certain properties after the removal of vertices or edges. In this paper, we will compare different measures of vulnerability with tenacity for several classes of graphs.