For a graph \( G = (V, E) \), a non-empty set \( S \subseteq V \) is a global offensive alliance (respectively, global strong offensive alliance) if for every vertex \( v \in V – S \), at least half of the vertices in its closed neighborhood are in \( S \) (respectively, a strict majority of its closed neighborhood are in \( S \)). The global offensive alliance number \( \gamma_o(G) \) (respectively, global strong offensive alliance number \( \gamma_{\hat{o}}(G) \)) is the minimum cardinality of a global offensive alliance (respectively, global strong offensive alliance) of \( G \). In this paper, we determine an upper bound on each parameter for bipartite graphs without isolated vertices. More precisely, we show that \( \gamma_o(G) \leq \frac{n – \ell + s}{2} \) and \( \gamma_{\hat{o}}(G) \leq \frac{n + \ell}{2} \), where \( n \), \( \ell \), and \( s \) are the order, the number of leaves, and the support vertices of \( G \), respectively. Moreover, extremal trees attaining each bound are characterized.

There is a hypothesis that a non-self-centric radially-maximal graph of radius \( r \) has at least \( 3r – 1 \) vertices. Moreover, if it has exactly \( 3r – 1 \) vertices, then it is planar with minimum degree \( 1 \) and maximum degree \( 3 \). Using an enhanced exhaustive computer search, we prove this hypothesis for \( r = 4, 5 \).

A vertex set \( X \) of a simple graph is called OO-irredundant if for each \( v \in X \), \( N(v) – N(X – \{v\}) \neq \emptyset \). Basic results for maximal OO-irredundant sets of a graph are obtained.

A set of necessary conditions for the existence of a partition of \(\{1, \ldots, 2m – 1, L\}\) into differences \(d, d + 1, \ldots, d + m – 1\) is \((m, L) \equiv (0, 0), (1, d + 1), (2, 1), (3, d) \pmod{(4, 2)}\) and \(m \geq 2d – 2\) or \(m = 1\). If \(m = 2d – 2\) then \(L = 5d – 5\), if \(m = 2d – 1\) then \(4d – 2 \leq L \leq 6d – 4\) and if \(m \geq 2d\) then \(2m \leq L \leq 3m + d – 2\). Similar conditions for the partition of \(\{1, \ldots, 2m, L\} \setminus \{2\}\) into differences \(d, d + 1, \ldots, d + m – 1\) are \((m, L) \equiv (0, 0), (1, d + 1), (2, 1), (3, d) \pmod{(4, 2)}\), \((d, m, L) \neq (1, 1, 4), (2, 3, 8)\) and \(m \geq 2d – 2\), \(m = 1\) or \((d, m, L) = (3, 2, 7), (3, 3, 9)\). If \(m = 2d – 2\) then \(L = 5d – 5, 5d – 3\), if \(m = 2d – 1\) then \(4d – 1 \leq L \leq 6d – 2\) and if \(m \geq 2d\) then \(2m + 1 \leq L \leq 3m + d – 1\).

It is shown that for many cases when the necessary conditions hold, the set \(\{1, \ldots, 2m – 1, L\}\) and \(\{1, \ldots, 2m – 1, L\} \setminus \{2\}\) can be so partitioned. These partitions exist for all the minimum and maximum \(L\), when \(d \leq 3\), when \(m = 1\) and when \(m \geq 8d – 3\) (\(m \geq 8d + 4\) in the hooked case). The constructions given fully solve the existence of these partitions if the necessary conditions for the existence of extended and hooked extended Langford sequences are sufficient.

Let \( n \) and \( q \) be positive integers, with \( n \geq 3 \), and let \( N = nq \). As input, we are to be given an arbitrary ordered \( n \)-sequence \( x_1, x_2, \ldots, x_n \), where \( 1 \leq x_i \leq N \) for all \( i \). We are to be presented with this sequence one entry at a time. As each entry is received, it must be placed into one of the positions \( 1, 2, \ldots, n \), where it must remain. A natural way to do this, in an attempt to sort the input sequence, is as follows. For any integer \( x \in \{1, \ldots, N\} \), we let \( s(x) \) denote the unique integer \( s \) for which \( (s-1)q + 1 \leq x \leq sq \). When we receive the entry \( x_i \), we consider those positions still unoccupied after having placed the previous \( i-1 \) entries, and we place \( x_i \) into the one which is closest to \( s(x_i) \). In the event of a tie for closest, we choose the higher of the two positions. We refer to this procedure as the \({placement\; algorithm}\). Regarding this algorithm, we consider the following question: for how many input sequences will the last two positions filled be positions \( 1 \) and \( n \)? We show that this number is \( (n-1)^{n-3}n^2q^n \).

The complexity of the maximum leaf spanning tree problem for grid graphs is currently unknown. We determine the maximum number of leaves in a grid graph with up to \(4\) rows and with \(6\) rows. Furthermore, we give some constructions of spanning trees of grid graphs with a large number of leaves.