In this paper, we determine the upper and lower bound for the total domination number and exact values and the upper bound for the double-total domination number on hexagonal grid H m , n with m hexagons in a row and n hexagons in a column. Further, we explore the ratio between the total domination number and the number of vertices of H m , n when m and n tend to infinity.

Double Roman domination is a stronger version of Roman domination that doubles the protection. The areas now have 0, 1, 2 or 3 legions. Every attacked area needs 2 legions for its defence, either their own, or borrowed from 1 or 2 neighbouring areas, which still have to keep at least 1 legion to themselves. The minimal number of legions in all areas together is equal to the double Roman domination number. In this paper we determine an upper bound and a lower bound for double Roman domination numbers on cardinal product of any two graphs. Also we determine the exact values of double Roman domination numbers on P 2 × G (for many types of graph G). Also, the double Roman domination number is found for P 2 × P n , P 3 × P n , P 4 × P n , while upper and lower bounds are given for P 5 × P n and P 6 × P n. Finally, we will give a case study to determine the efficiency of double protection. We will compare double Roman domination versus Roman domination by running a simulation of a battle.

This paper deals with the maximum weighted independent set (MWIS) problem. We consider several robust variants of the MWIS problem on trees and prove that most of them are NP-hard. We propose a heuristic for solving the considered robust MWIS variants, which is customized for trees. We demonstrate by experiments that our algorithm produces high-quality solutions and runs much faster than a general-purpose optimization software.

The maximum weighted independent set (MWIS) problem is important since it occurs in various applications, such as facility location, selection of non-overlapping time slots, labeling of digital maps, etc. However, in real-life situations, input parameters within those models are often loosely defined or subject to change. For such reasons, this paper studies robust variants of the MWIS problem. The study is restricted to cases where the involved graph is a tree. Uncertainty of vertex weights is represented by intervals. First, it is observed that the max–min variant of the problem can be solved in linear time. Next, as the most important original contribution, it is proved that the min–max regret variant is NP-hard. Finally, two mutually related approximation algorithms for the min–max regret variant are proposed. The first of them is already known, but adjusted to the considered situation, while the second one is completely new. Both algorithms are analyzed and evaluated experimentally.

This work deals with the robust maximum weighted independent set problem, i.e. finding a subset of graph vertices that are not adjacent to each other and whose sum of weights is as large as possible. Uncertainty in problem formulation is restricted to vertex weights and expressed explicitly by a finite set of scenarios. Three criteria of robustness are considered: absolute robustness (max-min), robust deviation (min-max regret), and relative robustness (relative min-max regret). Since the conventional maximum weighted independent set problem is already NP-hard, finding the exact solution of its robust counterpart should obviously have a prohibitive computational complexity. Therefore, we propose an approximate algorithm for solving the considered robust problem, which is based on evolutionary computing and on various crossover and mutation operators. The algorithm is experimentally evaluated on appropriate problem instances. It is shown that satisfactory solutions can be obtained for any of the three robustness criteria in reasonable time.