For some fixed \( n_0 \geq 0 \), we study the minimum number of vertices or edges that have to be removed from a graph such that no component of the rest has more than \( n_0 \) vertices.

An \( [r, s, n, t] \)-configuration is a collection \(C\) of \(r\)-sets in \( \{1, \ldots, n\} \) such that every \( s \)-set in \( \{1, \ldots, n\} \) contains at most \( t \) of the \( r \)-sets in \( C \). Studying this generalization of the Steiner system was suggested by a theorem of Poonen on union-closed families of sets. In this paper, we consider only \( [3, 4, n, 2] \)-configurations, and refer to them as \(n\)-configurations; by an \( (n, k) \)-configuration we mean an \(n\)-configuration containing exactly \(k\) \(3\)-sets. An \((n,k)\)-configuration is maximal if it is not contained in any \( (n, k + 1) \)-configuration; finally, \( L(n) \) is the largest integer \(k\) for which an \((n, k)\)-configuration exists. In this paper, we determine \(L(n)\) for \( 4 \leq n \leq 9 \), and characterize all the maximal \( n \)-configurations for \(n = 4, 5,\) and \(6\), as well as the \((n, L(n))\)-configurations for \( n = 7, 8, \) and \( 9 \).

Let \( G \) be a simple graph with vertex set \( V \) and edge set \( E \). A vertex labeling \( f: V \to \{0,1,2\} \) induces an edge labeling \( \bar{f}: E \to \{0,1,2\} \) defined by \( \bar{f}(uv) = |f(u) – f(v)| \). Let \( u_f(i) \) denote the number of vertices \( v \) with \( f(v) = i \), \( i = 0,1,2 \). Similarly, \( e_f(i) \) denotes the number of edges \( uv \) with \( \bar{f}(uv) = i \), \( i = 0,1,2 \). A graph is said to be \( 3 \)-equitable if there exists a vertex labeling \( f \) such that \( |v_f(i) – v_f(j)| \leq 1 \) and \( |e_f(i) – e_f(j)| \leq 1 \) for all \( i \neq j \), \( i, j = 0,1,2 \). In which case, \( f \) is called a \( 3 \)-equitable labeling.

In this paper, we prove that the following graphs are three equitable: (1) Helm graph \( H_n \) (\( n \geq 4 \)), (2) A Flower graph \( FL_n \), (3) One point union \( H_n^{(k)} \) of \( k \)-copies of \( H_n \), \( k \geq 1 \), (4) One point union \( K_4^{(k)} \) of \( k \) copies of \( K_4 \), (5) A \( K_4 \)-snake of \( n \) blocks, each equal to \( K_4 \), (6) A \( C_t \)-snake of \( n \) blocks, \( t = 4,6 \) and \( t = 5 \) with \( n \) not congruent to \( 3 \) modulo \( 6 \).

A defensive alliance in a graph \( G = (V,E) \) is a set of vertices \( S \subseteq V \) satisfying the condition that every vertex \( v \in S \) has at most one more neighbor in \( V – S \) than it has in \( S \). Because of such an alliance, the vertices in \( S \), agreeing to mutually support each other, have the strength of numbers to be able to defend themselves from the vertices in \( V – S \). In this paper, we introduce this new concept, together with a variety of other kinds of alliances, and initiate the study of their mathematical properties.

The distance-\( k \) domination number of graph \( G \), \( \gamma_{\leq k}(G) \), is the cardinality of a smallest set of vertices, \( S \), such that every vertex not in \( S \) is no more than distance \( k \) from at least one vertex of \( S \). Carrington, Harary, and Haynes showed \( |V^0| \geq 2|V^+| \) where \( V^0 = \{u \in V: \gamma_{\leq 1}(G-v) = \gamma_{\leq 1}(G)\} \) and \( V^+ = \{v \in V: \gamma_{\leq 1}(G-v) > \gamma_{\leq 1}(G)\} \). This paper extends the result to distance-\( k \) domination, with the obvious change in definition of \( V^0 \) and \( V^+ \), to show \( |V^0| \geq \frac{2}{2k-1}|V^+| \). Extremal graphs are characterized when \( k = 1 \) and some progress is mentioned on the characterization problem when \( k > 1 \).

We investigate constraints in finite Boolean lattices \( \langle \mathcal{P}(X), \subseteq \rangle \) where \( X \) is a finite set. The constraints studied here are of the form \( \langle Z,k \rangle \) where \( Z \subseteq X \), \( 1 \leq k \leq |Z| \). A set \( I \subseteq X \) \({satisfies}\) \( \langle Z,k \rangle \) if \( |I \cap Z| \geq k \). We characterize the sets satisfying collections of such constraints as filters (final segments) in \( \langle \mathcal{P}(X) \rangle \). We find yet other characterizations of filters including one by means of families of sets indexed by elements of \( X \) so that the elements of the filter correspond to subfamilies with an empty intersection. Our characterizations are supported for algorithms. We also study the families of negated constraints and mixed families and find their characterizations. In the positive case, formulas built of constraints can be used to measure the complexity of filters (and thus also of antichains of their minimal elements). We find pathological filters with very simple descriptions when the disjunctions are allowed, but extremely complex descriptions when only conjunctions are allowed.

The number \( g_3^{(4)}(v) \) represents the minimum cardinality of a pairwise balanced design on \( v \) elements in which the largest block size is four and every pair occurs exactly three times. We give a survey of the results for this quantity.