For each vertex \( v \) in a graph \( G \), let there be associated a particular type of a subgraph \( F_v \) of \( G \). In this context, the vertex \( v \) is said to dominate \( F_v \). A set \( S \) of vertices of \( G \) is called a full dominating set if every vertex of \( G \) belongs to a subgraph \( F_v \) of \( G \) for some \( v \in S \) and every edge of \( G \) belongs to a subgraph \( F_w \) of \( G \) for some \( w \in S \). The minimum cardinality of a full dominating set of \( G \) is its full domination number \( \gamma_F(G) \). A full dominating set of \( G \) of cardinality \( \gamma_F(G) \) is called a \( \gamma_F \)-set of \( G \).

We study three types of full domination in graphs: full star domination, where \( F_v \) is the maximum star centered at \( v \); full closed domination, where \( F_v \) is the subgraph induced by the closed neighborhood of \( v \); and full open domination, where \( F_v \) is the subgraph induced by the open neighborhood of \( v \).

A subset \( T \) of a \( \gamma_F \)-set \( S \) in a graph \( G \) is a forcing subset for \( S \) if \( S \) is the unique \( \gamma_F \)-set containing \( T \). The forcing full domination number of \( S \) in \( G \) is the minimum cardinality of a forcing subset for \( S \), and the forcing full domination number \( f_{\gamma_F}(G) \) of the graph \( G \) is the minimum forcing full domination number among all \( \gamma_F \)-sets of \( G \).

We present several realization results concerning forcing parameters in full domination.

A minimum feedback arc set of a digraph is a smallest sized set of arcs whose reversal makes the resulting digraph acyclic. Given an acyclic digraph \( D \), we seek a smallest sized tournament \( T \) having \( A(D) \) as a minimum feedback arc set. The reversing number of a digraph \( D \) equals \( |V(T)| – |V(D)| \). We investigate the reversing number of the \( k \)th power of directed Hamiltonian path \( P_n^k \), when \( k \) is fixed and \( n \) tends to infinity. We show that even for small values of \( k \), where \( |A(P_n^k)| \) is much closer to \( |A(P_n)| \) than \( |A(T_n)| \), the opposite relationship holds for the reversing number.

A union closed (UC) family \( \mathcal{A} \) is a finite family of sets such that the union of any two sets in \( \mathcal{A} \) is also in \( \mathcal{A} \). Peter Frankl conjectured in 1979 that for every union closed family \( \mathcal{A} \), there exists some \( x \) contained in at least half the members of \( \mathcal{A} \). In this paper, we show that if a UC family \( \mathcal{A} \) fails the conjecture, then no element can appear in more than two of its \( 3 \)-sets, and so the number of \( 3 \)-sets in \( \mathcal{A} \) can be no more than \( \frac{2n}{3} \).

A \( (k,d) \)-total coloring (\( k,d \in \mathbb{N}, k \geq 2d \)) of a graph \( G \) is an assignment \( c \) of colors \( \{0,1,\ldots,k-1\} \) to the vertices and edges of \( G \) such that \( d \leq |c(x_i) – c(x_j)| \leq k – d \) whenever \( x_i \) and \( x_j \) are two adjacent edges, two adjacent vertices, or an edge incident to a vertex. The circular total chromatic number \( \chi_c”(G) \) is defined by \(\chi_c”(G) = \inf\{k/d : G \text{ has a } (k, d)\text{-total coloring}\}.\) It was proved that \( \chi”(G) – 1 < \chi_c''(G) \leq \chi''(G) \) — where \( \chi''(G) \) is the total chromatic number of \( G \) — with equality for all type-1 graphs and most of the so far considered type-2 graphs. We determine an infinite class of graphs \( G \) such that \( \chi_c''(G) < \chi''(G) \) and we list all graphs of order \( <7 \) with this property.

In this paper we consider a variation of the classical Turán-type extremal problems. Let \( S \) be an \( n \)-term graphical sequence, and \( \sigma(S) \) be the sum of the terms in \( S \). Let \( H \) be a graph. The problem is to determine the smallest even \( l \) such that any \( n \)-term graphical sequence \( S \) having \( \sigma(S) \geq l \) has a realization containing \( H \) as a subgraph. Denote this value \( l \) by \( \sigma(H, n) \). We show \(\sigma(C_{2m+1}, n) = m(2n – m – 1) + 2, \quad \text{for } m \geq 3, n \geq 3m;\) \(\sigma(C_{2m+2}, n) = m(2n – m – 1) + 4, \quad \text{for } m \geq 3, n \geq 5m – 2. \)

We first prove that if \( G \) is a connected graph with \( n \) vertices and chromatic number \( \chi(G) = k \geq 2 \), then its independent domination number

\[i(G) \leq \left\lceil \frac{(k-1)}{k}n \right\rceil – (k-2).\]

This bound is tight and remains so for planar graphs. We then prove that the independent domination number of a diameter two planar graph on \( n \) vertices is at most \( \left\lceil \frac{n}{3} \right\rceil \).

Hill, Landjev, Jones, Storme, and Barat proved in a previous article on caps in \(PG(5, 3)\) and \(PG(6,3)\) that every 53-cap in \(PG(5, 3)\) is contained in the 56-cap of Hill and that there exist complete 48-caps in \(PG(5,3)\). The first result was used to lower the upper bound on \( m_2(6,3) \) on the size of caps in \(PG(6, 3)\) from 164 to 154. Presently, the known upper bound on \( m_2(6, 3) \) is 148. In this article, using computer searches, we prove that every 49-cap in \(PG(5, 3)\) is contained in a 56-cap, and that every 48-cap, having a 20-hyperplane with at most 8-solids, is also contained in a 56-cap. Computer searches for caps in \(PG(6,3)\) which use the computer results of \(PG(5,3)\) then lower the upper bound on \( m_2(6,3) \) to \( m_2(6,3) \leq 136 \). So now we know that \( 112 \leq m_2(6,3) \leq 136 \).

Let \( \delta(G) \) and \( \lambda(G) \) be the minimum degree and edge-connectivity of a graph \( G \), respectively. A graph \( G \) is maximally edge-connected if \( \lambda(G) = \delta(G) \) and super-edge-connected if every minimum edge cut consists of edges adjacent to a vertex of minimum degree.

In this paper, sufficient conditions for super-edge-connected graphs depending on the clique number and the minimum degree are presented. These results show that some known sufficient conditions for maximally edge-connected graphs even lead to super-edge-connected graphs.

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 \( 9 \).