This paper is devoted to the study of quadruple Roman domination in trees, and it a contribution Special Issue “Theoretical computer science discrete mathematics” Symmetry. For any positive integer k, [k]-Roman dominating function ([k]-RDF) simple graph G from vertex set V {0,1,2,…,k+1} if for u?V with f(u)<k, ?x?N(u)?{u}f(x)?|{x?N(u):f(x)?1}|+k, where N(u) open neighborhood u. The weight [k...

‎for any integer $kgeq 1$‎, ‎a set $s$ of vertices in a graph $g=(v,e)$ is a $k$-‎tuple total dominating set of $g$ if any vertex‎ ‎of $g$ is adjacent to at least $k$ vertices in $s$‎, ‎and any vertex‎ ‎of $v-s$ is adjacent to at least $k$ vertices in $v-s$‎. ‎the minimum number of vertices of such a set‎ ‎in $g$ we call the $k$-tuple total restrained domination number of $g$‎. ‎the maximum num...

Let K n denote the Cartesian product Kn Kn Kn, where Kn is the complete graph on n vertices. We show that the domination number of K n is ⌈

Domination theory is a well-established topic in graph theory, as well one of the most active research areas. Interest this area partly explained by its diversity applications to real-world problems, such facility location computer and social networks, monitoring communication, coding algorithm design, among others. In last two decades, functions defined on graphs have attracted attention sever...

We analyze the graph-theoretic formalization of Roman domination, dating back to the military strategy of Emperor Constantine, from a parameterized perspective. More specifically, we prove that this problem is W[2]-complete for general graphs. However, parameterized algorithms are presented for graphs of bounded treewidth and for planar graphs. Moreover, it is shown that a parametric dual of Ro...

This paper is motivated by the concept of nonnegative signed domination that was introduced by Huang, Li, and Feng in 2013 [15]. We study the non-negative signed domination problem from the theoretical point of view. For networks modeled by strongly chordal graphs and distance-hereditary graphs, we show that the non-negative signed domination problem can be solved in polynomial time. For networ...