The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for $k\ge 3$. In this paper we introduce five new heuristics. Two algorithms, ${\mathrm{B}}_m$ and ${\mathrm{C}}_m$, arise as natural improvements of Algorithm ${\mathrm{A}}_m$ from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004). The other three algorithms, ${\mathrm{D}}_m$, ${\mathrm{E}}_m$, and ${\mathrm{F}}_m$, incorporate randomization. Algorithm ${\mathrm{D}}_m$ can be considered as a greedy version of ${\mathrm{B}}_m$, whereas ${\mathrm{E}}_m$ and ${\mathrm{F}}_m$ are versions of local search algorithm, specialized for the k-matching problem. The algorithms are implemented in Python and are run on three datasets. On the datasets available, all the algorithms clearly outperform Algorithm ${\mathrm{A}}_m$ in terms of solution quality. On the first dataset with known optimal values the average relative error ranges from 1.47% over optimum (algorithm ${\mathrm{A}}_m$) to 0.08% over optimum (algorithm ${\mathrm{E}}_m$). On the second dataset with known optimal values the average relative error ranges from 4.41% over optimum (algorithm ${\mathrm{A}}_m$) to 0.45% over optimum (algorithm ${\mathrm{F}}_m$). Better quality of solutions demands higher computation times, thus the new algorithms provide a good compromise between quality of solutions and computation time.

Since 2016, when the first paper of the double Roman domination appeared, the topic has received considerable attention in the literature. We survey known results on double Roman domination and some variations of the double Roman domination, and a list of open questions and conjectures is provided.

Closeness is a measure of centrality, an important feature of communication and social networks. Extremal networks among all graphs and among several subclasses of graphs including trees and cacti are given. In addition, maximal graphs among cacti with fixed number of cycles and among cacti with given number of cut edges are provided.

We obtain new results on the 2-rainbow domination number of generalized Petersen graphs $P(ck,k)$. Exact values are established for all infinite families where the general lower bound $\frac{4}{5}ck$ is attained. In all other cases lower and upper bounds with small gaps are given.