Let \( k \) be a positive integer and \( G = (V, E) \) be a graph of minimum degree at least \( k – 1 \). A function \( f: V \to \{-1, 1\} \) is called a \({signed \; k -dominating\; function}\) of \( G \) if \( \sum_{u \in N_G[v]} f(u) \geq k \) for all \( v \in V \). The \({signed \; k -domination \;number}\) of \( G \) is the minimum value of \( \sum_{v \in V} f(v) \) taken over all signed \( k \)-dominating functions of \( G \). The \({signed \;total \; k-dominating \;function}\) and \({signed\; total \; k -domination\; number}\) of \( G \) can be similarly defined by changing the closed neighborhood \( N_G[v] \) to the open neighborhood \( N_G(v) \) in the definition. The upper \({signed \; k -domination \;number}\) is the maximum value of \( \sum_{u \in V} f(u) \) taken over all \({minimal}\) signed \( k \)-dominating functions of \( G \). In this paper, we study these graph parameters from both algorithmic complexity and graph-theoretic perspectives. We prove that for every fixed \( k \geq 1 \), the problems of computing these three parameters are all \( \mathcal{NP} \)-hard. We also present sharp lower bounds on the signed \( k \)-domination number and signed total \( k \)-domination number for general graphs in terms of their minimum and maximum degrees, generalizing several known results about signed domination.

In this paper, we study a pair of simplicial complexes, which we denote by \( \mathcal{B}(k,d) \) and \( \mathcal{ST}(k+1,d-k-1) \), for all nonnegative integers \( k \) and \( d \) with \( 0 \leq k \leq d-2 \). We conjecture that their underlying topological spaces \( |\mathcal{B}(k,d)| \) and \( |\mathcal{ST}(k+1,d-k-1)| \) are homeomorphic for all such \( k \) and \( d \). We answer this question when \( k = d-2 \) by relating the complexes through a series of well-studied combinatorial operations that transform a combinatorial manifold while preserving its PL-homeomorphism type.

Let \( D = (V,A) \) be a finite and simple digraph. A Roman dominating function (RDF) on \( D \) is a labeling \( f: V(D) \to \{0,1,2\} \) such that every vertex \( v \) with label \( 0 \) has a vertex \( w \) with label \( 2 \) such that \( wv \) is an arc in \( D \). The weight of an RDF \( f \) is the value \( \omega(f) = \sum_{v \in V} f(v) \). The Roman domination number of a digraph \( D \), denoted by \( \gamma_R(D) \), equals the minimum weight of an RDF on \( D \). The Roman reinforcement number \( r_R(D) \) of a digraph \( D \) is the minimum number of arcs that must be added to \( D \) in order to decrease the Roman domination number. In this paper, we initiate the study of Roman reinforcement number in digraphs and we present some sharp bounds for \( r_R(D) \). In particular, we determine the Roman reinforcement number of some classes of digraphs.

In this paper, some formulae for computing the numbers of spanning trees of the corona and the join of graphs are deduced.

Partially filled \(6 \times 6\) Sudoku grids are categorized based on the arrangement of the values in the first three rows. This categorization is then employed to determine the number of \(6 \times 6\) Sudoku grids.

Stankova and West proved in 2002 that the patterns \( 231 \) and \( 312 \) are shape-Wilf-equivalent. Their proof was nonbijective. We give a new characterization of \( 231 \) and \( 312 \) avoiding full rook placements and use this to give a simple bijection that demonstrates the shape-Wilf-equivalence.