Let \(\tau(G)\) denote the number of vertices in a longest path of the graph \(G = (V, E)\). A subset \(K\) of \(V\) is called a \(P_n\)-\emph{kernel} of \(G\) if \(\tau(G[K]) \leq n – 1\) and every vertex \(v \in V(G – K)\) is adjacent to an end-vertex of a path of order \(n – 1\) in \(G[K]\).
A partition \(\{A, B\}\) of \(V\) is called an \((a, b)\)-partition if \(\tau(G[A]) \leq a\) and \(\tau(G[B]) \leq b\).
We show that any graph with girth greater than \(n – 3\) has a \(P_n\)-kernel and that every graph has a \(P_\gamma\)-kernel. As corollaries of these results, we show that if \(\tau(G) = a + b\) and \(G\) has girth greater than \(a – 2\) or \(a \leq 6\), then \(G\) has an \((a, b)\)-partition.

In this paper, we establish that for arbitrary positive integers k and m, where \(k > 1\), there exists a tournament which has exactly m minimum dominating sets of order k. A construction of such tournaments will be given.

A connected graph \(G\) is \((\gamma, k)\)-insensitive if the domination number \(\gamma(G)\) is unchanged when an arbitrary set of \(k\) edges is removed. The problem of finding the least number of edges in any such graph has been solved for \(k = 1\) and for \(k = \gamma(G) = 2\). Asymptotic results as \(n\) approaches infinity are known for \(k \geq 2\) and \(k+1 \leq \gamma(G) \leq 2k\). Note that for \(k = 2\), this bound holds only for graphs \(G$ with \(\gamma(G) \in \{3,4\}\). In this paper, we present an asymptotic bound for the minimum number of edges in an extremal \((\gamma, k)\)-insensitive graph \(G\), where \(k = 2\) and \(n \geq 3\gamma(G)^2 – 2\gamma(G) + 3\) that holds for \(\gamma(G) \geq 3\). For small \(n\), we present tighter bounds (in some cases exact values) for this minimum number of edges.

A queen on a hexagonal board with hexagonal cells is defined as a piece that moves along three lines, namely along the cells in the same row, up diagonal, or down diagonal. A queen dominates a cell if the cell is in the same line as the queen.
We show that hexagonal boards with \(n \geq 1\) rows and diagonals, where \(n \equiv 3 \pmod{4}\), have only two types of minimum dominating sets. We also determine the irredundance numbers of the boards with \(5\) and \(7\) rows.

A well-spread sequence is an increasing sequence of distinct positive integers whose pairwise sums are distinct. Some properties of these sequences are discussed.

In this note, we consider finite, undirected, and simple graphs. A subset \(D\) of the vertex set of a graph \(G\) is a dominating set if each vertex of \(G\) is either in \(D\) or adjacent to some vertex of \(D\). A dominating set of minimum cardinality is called a minimum dominating set
A vertex \(v\) of a graph \(G\) is called a cut-vertex of G if \(G – v\) has more components than \(G\). A block of a graph is a maximal connected subgraph having no cut-vertex.
A block-cactus graph is a graph whose blocks are either complete graphs or cycles, and we speak of a cactus if the complete graphs consist of only one edge.
In our main theorem, we shall show that the minimum dominating set problem of an arbitrary graph can be reduced to its blocks. This theorem provides a linear-time algorithm for determining a minimum dominating set in a block-cactus graph, and thus, it can be seen as a supplement to a linear-time algorithm for finding a minimum dominating set in a cactus, presented by S.T. Hedetniemi, R.C. Laskar, and J. Pfaff in 1986.

For a graph facility or multi-facility location problem, each vertex is typically considered to be the location for one customer or one facility. Typically, the number of facilities is predetermined, and one must optimally locate these facilities so as to minimize some function of the distances between customers and facilities (and, perhaps, of the distances among the facilities). For example, p facility locations (such as, for hospitals or fire stations) might be chosen so as to minimize the maximum or the average distance from a customer to the nearest facility. The problem investigated in this paper considers all of the facilities to be distinct, and we seek to minimize the average customer-to-facility distance, primarily for grid graphs.

Let \(G = (V, E)\) be a graph.A set \(S \subseteq V\) is a dominating set if every vertex not in \(S\) is adjacent to a vertex in \(S\).The domination number of \(G\), denoted by \(\gamma(G)\), is the minimum cardinality of a dominating set of \(G\).Sanchis [8] showed that a connected graph \(G\) of size \(q\) and minimum degree at least \(2\) has domination number at most \((q+2)/3\).
In this paper, connected graphs \(G\) of size \(q\) with minimum degree at least \(2\) satisfying \(\gamma(G) > \frac{q}{3}\) are characterized.

For two vertices \(u\) and \(v\) in a strong oriented graph \(D\) of order \(n \geq 3\), the strong distance \(\text{sd}(u,v)\) between \(u\) and \(v\) is the minimum size of a strong subdigraph of \(D\) containing \(u\) and \(v\). For a vertex \(v\) of \(D\), the strong eccentricity \(\text{se}(v)\) is the strong distance between \(v\) and a vertex farthest from \(v\). The minimum strong eccentricity among the vertices of \(D\) is the strong radius \(\text{srad}(D)\), and the maximum strong eccentricity is its strong diameter \(\text{sdiam}(D)\). It is shown that every pair \(r,d\) of integers with \(3 \leq r \leq d \leq 2r\) is realizable as the strong radius and strong diameter of some strong oriented graph.
Moreover, for every strong oriented graph \(D\) of order \(n \geq 3\), it is shown that
\[
\text{sdiam}(D) \leq \left\lfloor \frac{5(n-1)}{3} \right\rfloor.
\]
Furthermore, for every integer \(n \geq 3\), there exists a strong oriented graph \(D\) of order \(n\) such that
\[
\text{sdiam}(D) = \left\lfloor \frac{5(n-1)}{3} \right\rfloor.
\]

For each vertex \(s\) of the subset \(S\) of vertices of a graph \(G\), we define Boolean variables \(p\), \(q\), \(r\) which measure the existence of three kinds of \(S\)-private neighbors (\(S\)-pns) of \(s\). A \(3\)-variable Boolean function \(f = f(p, q, r)\) may be considered as a compound existence property of \(S\)-pns. The set \(S\) is called an \(f\)-set of \(G\) if \(f = 1\) for all \(s \in S\), and the class of all \(f\)-sets of \(G\) is denoted by \(\Omega_f\). Special cases of \(\Omega_f\) include the
independent sets, irredundant sets, open irredundant sets, and CO-irredundant sets of \(G\).
There are \(62\) non-trivial families \(\Omega_f\) which include the \(7\) families of a framework proposed earlier by Fellows, Fricke, Hedetniemi, and Jacobs. The functions \(f\) for which \(\Omega_f\) is hereditary for any graph \(G\) are determined. Additionally, the existence and properties of \(f\)-Ramsey numbers (analogues of the elusive classical Ramsey numbers) are investigated, and future directions for the theory of the classes \(\Omega_f\) are considered.