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The study of Roman domination has evolved to encompass a variety of challenging extensions, each contributing to the broader understanding of domination problems in graph theory. This paper explores the Perfect Location Signed Roman Domination (PLSRD) problem, a novel combination of the Perfect Roman, Locating Roman, and Signed Roman Domination paradigms. In PLSRD, each weak vertex, assigned the label -1, must be protected by exactly one strong vertex, with additional limitation that two weak vertices cannot share the same strong vertex, while the total sum of labels in the closed neighborhood of each vertex must remain positive. This paper provides exact values for the PLSRD number in several well-known graph classes, including complete graphs, complete bipartite graphs, wheels, paths, cycles, ladders, prism graphs, and 3 x n grids. Additionally, we establish a lower bound for a general 3 regular graph, as well as the upper bounds for flower snarks graphs, highlighting the intricate interplay between the PLSRD constraints and the structural properties of these graph families.

The concept of Roman domination has been a subject of intrigue for more than two decades with the fundamental Roman domination problem standing out as one of the most significant challenges in this field. This article studies a practically motivated generalization of this problem, known as the k-strong Roman domination. In this variation, defenders within a network are tasked with safeguarding any k vertices simultaneously, under multiple attacks. The objective is to find a feasible mapping that assigns an (integer) weight to each vertex of the input graph with a minimum sum of weights across all vertices. A function is considered feasible if any non-defended vertex, i.e. one labeled by zero, is protected by at least one of its neighboring vertices labeled by at least two. Furthermore, each defender ensures the safety of a non-defended vertex by imparting a value of one to it while always retaining a one for themselves. To the best of our knowledge, this paper represents the first theoretical study on this problem. The study presents results for general graphs, establishes connections between the problem at hand and other domination problems, and provides exact values and bounds for specific graph classes, including complete graphs, paths, cycles, complete bipartite graphs, grids, and a few selected classes of convex polytopes. Additionally, an attainable lower bound for general cubic graphs is provided.

This paper considers the following three Roman domination graph invariants on Kneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $\gamma_{R}(K_{n,k})$ and total Roman domination number $\gamma_{tR}(K_{n,k})$ proving that for $n\geqslant k(k+1)$, $\gamma_{R}(K_{n,k}) =\gamma_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $\gamma_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $n\geqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature.

In a network, a $k$-plex represents a subset of $n$ vertices where the degree of each vertex in the subnetwork induced by this subset is at least $n-k$. The maximum edge-weight $k$-plex partitioning problem (Max-EkPP) is to find the $k$-plex partitioning in edge-weighted network, such that the sum of edge weights is maximal. The Max-EkPP has an important role in discovering new information in large sparse biological networks. We propose a variable neighborhood search (VNS) algorithm for solving Max-EkPP. The VNS implements a local search based on the 1-swap first improvement strategy and the objective function that takes into account the degree of every vertex in each partition. The objective function favors feasible solutions, also enabling a gradual increase in terms of objective function value when moving from slightly infeasible to barely feasible solutions. A comprehensive experimental computation is performed on real metabolic networks and other benchmark instances from literature. Comparing to the integer linear programming method from literature, our approach succeeds to find all known optimal solutions. For all other instances, the VNS either reaches previous best known solution or improves it. The proposed VNS is also tested on a large-scaled dataset which was not previously considered in literature., Comment: Submitted to a scientific journal in November 2017