We define the \( (i, j) \)-liars’ domination number of \( G \), denoted by \( LR(i, j)(G) \), to be the minimum cardinality of a set \( L \subseteq V(G) \) such that detection devices placed at the vertices in \( L \) can precisely determine the set of intruder locations when there are between 1 and \( i \) intruders and at most \( j \) detection devices that might “lie”.

We also define the \( X(c_1, c_2, \ldots, c_t, \ldots) \)-domination number, denoted by \( \gamma _{X(c_1, c_2, \ldots, c_t, \ldots)}(G) \), to be the minimum cardinality of a set \( D \subseteq V(G) \) such that, if \( S \subseteq V(G) \) with \( |S| = k \), then \( |(\bigcup_{v \in S} N[v]) \cap D| \geq c_k \). Thus, \( D \) dominates each set of \( k \) vertices at least \( c_k \) times making \( \gamma_{X(c_1, c_2, \ldots, c_t, \ldots)}(G) \) a set-sized dominating parameter. We consider the relations between these set-sized dominating parameters and the liars’ dominating parameters.

For Cauchy numbers of the first kind \( \{a_n\}{n \geq 0} \) and Cauchy numbers of the second kind \( \{b_n\}{n \geq 0} \), this paper focuses on the log-convexity of some sequences related to \( \{a_n\}{n \geq 0} \) and \( \{b_n\}{n \geq 0} \). For example, we discuss log-convexity of \( \{n|a_n| – |a_{n+1}|\}{n \geq 1} \), \( \{b{n+1} – nb_n\}{n \geq 1} \), \( \{n|a_n|\}{n \geq 1} \), and \( \{(n + 1)b_n\}_{n \geq 0} \). In addition, we investigate log-balancedness of some sequences involving \( a_n \) (or \( b_n \)).

Let \( G \) be a graph. We define the distance \( d \) pebbling number of \( G \) to be the smallest number \( s \) such that if \( s \) pebbles are placed on the vertices of \( G \), then there must exist a sequence of pebbling moves which takes a pebble to a vertex which is at a distance of at least \( d \) from its starting point. In this article, we evaluate the distance \( d \) pebbling numbers for a directed cycle graph with \( n \) vertices.

Let \( G \) be a \( k \)-connected (\( k \geq 2 \)) graph of order \( n \). If \( \gamma(G^c) \geq n – k \), then \( G \) is Hamiltonian or \( K_k \vee K_{k+1}^c \), where \( \gamma(G^c) \) is the domination number of the complement of the graph \( G \).

An \emph{Italian dominating function} on a digraph \( D \) with vertex set \( V(D) \) is defined as a function \( f : V(D) \to \{0, 1, 2\} \) such that every vertex \( v \in V(D) \) with \( f(v) = 0 \) has at least two in-neighbors assigned 1 under \( f \) or one in-neighbor \( w \) with \( f(w) = 2 \). The weight of an Italian dominating function is the sum \( \sum_{v \in V(D)} f(v) \), and the minimum weight of an Italian dominating function \( f \) is the \emph{Italian domination number}, denoted by \( \gamma_I(D) \). We initiate the study of the Italian domination number for digraphs, and we present different sharp bounds on \( \gamma_I(D) \). In addition, we determine the Italian domination number of some classes of digraphs. As applications of the bounds and properties on the Italian domination number in digraphs, we give some new and some known results of the Italian domination number in graphs.