Let \( G \) be a simple graph, and let \( p \) be a positive integer. A subset \( D \subseteq V(G) \) is a \( p \)-\({dominating}\) set of the graph \( G \), if every vertex \( v \in V(G) – D \) is adjacent to at least \( p \) vertices of \( D \). The \( p \)-domination number \( \gamma_p(G) \) is the minimum cardinality among the \( p \)-dominating sets of \( G \). Note that the \( 1 \)-domination number \( \gamma_1(G) \) is the usual domination number \( \gamma(G) \). The covering number of a graph \( G \) is denoted by \( \beta(G) \). If \( T \) is a tree of order \( n(T) \), then Fink and Jacobson [1] proved in 1985 that

\[\gamma_p(T) \geq \frac{(p-1)n(T) + 1}{p}\]

The special case \( p = 2 \) of this inequality easily leads to

\[\gamma_2(T) \geq \beta(T) + 1 \geq \gamma(T) + 1\]

for every non-trivial tree \( T \). Inspired by the article of Fink and Jacobson [1], we characterize in this paper the family of trees \( T \) with \( \gamma_p(T) = \left\lceil \frac{(p-1)n(T) + 1}{p} \right\rceil \) as well as all non-trivial trees \( T \) with \( \gamma_2(T) = \gamma(T) + 1 \) and \( \gamma_2(T) = \beta(T) + 1 \).

Alliances in undirected graphs were introduced by Hedetniemi, Hedetniemi, and Kristiansen, and generalized to \( k \)-alliances by Shafique and Dutton. We translate these definitions of alliances to directed graphs. We establish basic properties of alliances and examine bounds on the size of minimal alliances in directed graphs. In general, the bounds established for alliances in undirected graphs do not hold when alliances are considered over the larger class of directed graphs and we construct examples which break these bounds.

For given integers \( k \) and \( \ell \), \( 3 \leq k \leq \ell \), a graphic sequence \( \pi = (d_1, d_2, \dots, d_n) \) is said to be potentially \({}_{k}C_\ell\)-graphic if there exists a realization of \( \pi \) containing \( C_r \), for each \( r \), where \( k \leq r \leq \ell \) and \( C_r \) is the cycle of length \( r \). Luo (Ars Combinatoria 64(2002)301-318) characterized the potentially \( C_\ell \)-graphic sequences without zero terms for \( r = 3, 4, 5 \). In this paper, we characterize the potentially \({}_{k}C_\ell\)-graphic sequences without zero terms for \( k = 3, 4 \leq \ell \leq 5 \) and \( k = 4, \ell = 5 \).

We show that deciding if a set of vertices is an eternal \(1\)-secure set is complete for \(\text{co-}NP^{\text{NP}}\), solving a problem stated by Goddard, Hedetniemi, and Hedetniemi \([JCMCC, \text{vol. 52}, \text{pp. 160-180}]\).

A Sarvate-Beam type of triple system is defined in the case \( v \equiv 2 \pmod{3} \) and an enumeration is given of such systems for \( v = 5 \).

Informally, a set of guards positioned on the vertices of a graph \( G \) is called eternally secure if the guards are able to respond to vertex attacks by moving a single guard along a single edge after each attack regardless of how many attacks are made. The smallest number of guards required to achieve eternal security is the eternal security number of \( G \), denoted \( es(G) \), and it is known to be no more than \( \theta_v(G) \), the vertex clique cover number of \( G \). We investigate conditions under which \( es(G) = \theta_v(G) \).

We apply Computational Algebra methods to the construction of Hadamard matrices from two circulant submatrices, given by C. H. Yang. We associate Hadamard ideals to this construction, to systematize the application of Computational Algebra methods. Our approach yields an exhaustive search for Hadamard matrices from two circulant submatrices for this construction, for the first eight admissible values \(2, 4, 8, 10, 16, 18, 20, 26\) and partial searches for the next three admissible values \(32, 34, 40\). From the solutions we found, for the admissible values \(26\) and \(34\), we located new inequivalent Hadamard matrices of orders \(52\) and \(68\) with two circulant submatrices, thus improving the lower bounds for the numbers of inequivalent Hadamard matrices of orders \(52\) and \(68\). We also propose a heuristic decoupling of one of the equations arising from this construction, which can be used together with the PSD test to search for solutions more efficiently.

A Hamilton cycle in an \( n \)-cube is said to be \( k \)-warped if its \( k \)-paths have their edges running along different parallel \( 1 \)-factors. No Hamilton cycle in the \( n \)-cube can be \( n \)-warped. The equivalence classes of Hamilton cycles in the \( 5 \)-cube are represented by the circuits associated to their corresponding minimum change-number sequences, or minimum \( H \)-circuits. This makes feasible an exhaustive search of such Hamilton cycles allowing their classification according to class cardinalities, distribution of change numbers, duplicity, reversibility, and \( k \)-warped representability, for different values of \( k < n \). This classification boils down to a detailed enumeration of a total of \( 237675 \) equivalence classes of Hamilton cycles in the \( 5 \)-cube, exactly four of which do not traverse any sub-cube. One of these four classes is the unique class of \( 4 \)-warped Hamilton cycles in the \( 5 \)-cube. In contrast, there is no \( 5 \)-warped Hamilton cycle in the \( 6 \)-cube. On the other hand, there is exactly one class of Hamilton cycles in the graph of middle levels of the \( 5 \)-cube. A representative of this class possesses an elegant geometrical and symmetrical disposition inside the \( 5 \)-cube.

The main objective of this paper is to introduce a generalization of distance called superior distance in graphs. For two vertices \( u \) and \( v \) of a connected graph, we define \( \text{D}_{u,v} = \text{N}[u] \cup \text{N}[v] \). We define a \( \text{D}_{u,v} \)-walk as a \( u \)-\( v \) walk that contains every vertex of \( \text{D}_{u,v} \). The superior distance \( \text{d}_D(u,v) \) from \( u \) to \( v \) is the length of a shortest \( \text{D}_{u,v} \)-walk. In this paper, first we give the bounds for the superior diameter of a graph and a property that relates the superior eccentricities of adjacent vertices. Finally, we investigate those graphs that are isomorphic to the superior center of some connected graph and those graphs that are isomorphic to the superior periphery of some connected graph.

For any \( h \in \mathbb{N} \), a graph \( G = (V, E) \) is said to be \( h \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_h – \{0\} \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_h \) defined by

\[l^+(v) = \sum_{uv \in E(G)} l(uv)\]

is a constant map. For a given graph \( G \), the set of all \( h \in \mathbb{Z}_+ \) for which \( G \) is \( h \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). The concept of integer-magic spectrum of a graph was first introduced in [4]. But unfortunately, this paper has a number of incorrect statements and theorems. In this paper, first we will correct some of those statements, then we will determine the integer-magic spectra of caterpillars.