A Survey on Alliances and Related Parameters in Graphs

In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global $r$-alliances in graphs.We also propose a new framework, called (global) $(D,O)$-alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations

Signed double Roman domination on cubic graphs

The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from $\{\pm{}1,2,3\}$ to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility is given whenever (i) vertices labeled $\pm{}1$ have at least one neighbor with label in $\{2,3\}$; (ii) each vertex labeled $-1$ has one $3$-labeled neighbor or at least two $2$-labeled neighbors; and (iii) the sum of labels over the closed neighborhood of any vertex is positive. The cumulative weight of an optimal labeling is called signed double Roman domination number (SDRDN). In this work, we first consider the problem on general cubic graphs of order $n$ for which we present a sharp $n/2+\Theta(1)$ lower bound for the SDRDN by means of the discharging method. Moreover, we derive a new best upper bound. Observing that we are often able to minimize the SDRDN over the class of cubic graphs of a fixed order, we then study in this context generalized Petersen graphs for independent interest, for which we propose a constraint programming guided proof. We then use these insights to determine the SDRDNs of subcubic $2\times m$ grid graphs, among other results

A Hierachical Evolutionary Algorithm for Multiobjective Optimization in IMRT

Purpose: Current inverse planning methods for IMRT are limited because they are not designed to explore the trade-offs between the competing objectives between the tumor and normal tissues. Our goal was to develop an efficient multiobjective optimization algorithm that was flexible enough to handle any form of objective function and that resulted in a set of Pareto optimal plans. Methods: We developed a hierarchical evolutionary multiobjective algorithm designed to quickly generate a diverse Pareto optimal set of IMRT plans that meet all clinical constraints and reflect the trade-offs in the plans. The top level of the hierarchical algorithm is a multiobjective evolutionary algorithm (MOEA). The genes of the individuals generated in the MOEA are the parameters that define the penalty function minimized during an accelerated deterministic IMRT optimization that represents the bottom level of the hierarchy. The MOEA incorporates clinical criteria to restrict the search space through protocol objectives and then uses Pareto optimality among the fitness objectives to select individuals. Results: Acceleration techniques implemented on both levels of the hierarchical algorithm resulted in short, practical runtimes for optimizations. The MOEA improvements were evaluated for example prostate cases with one target and two OARs. The modified MOEA dominated 11.3% of plans using a standard genetic algorithm package. By implementing domination advantage and protocol objectives, small diverse populations of clinically acceptable plans that were only dominated 0.2% by the Pareto front could be generated in a fraction of an hour. Conclusions: Our MOEA produces a diverse Pareto optimal set of plans that meet all dosimetric protocol criteria in a feasible amount of time. It optimizes not only beamlet intensities but also objective function parameters on a patient-specific basis

The Power of Small Coalitions under Two-Tier Majority on Regular Graphs

In this paper, we study the following problem. Consider a setting where a proposal is offered to the vertices of a given network $G$, and the vertices must conduct a vote and decide whether to accept the proposal or reject it. Each vertex $v$ has its own valuation of the proposal; we say that $v$ is ``happy'' if its valuation is positive (i.e., it expects to gain from adopting the proposal) and ``sad'' if its valuation is negative. However, vertices do not base their vote merely on their own valuation. Rather, a vertex $v$ is a \emph{proponent} of the proposal if the majority of its neighbors are happy with it and an \emph{opponent} in the opposite case. At the end of the vote, the network collectively accepts the proposal whenever the majority of its vertices are proponents. We study this problem for regular graphs with loops. Specifically, we consider the class $\mathcal{G}_{n|d|h}$ of $d$-regular graphs of odd order $n$ with all $n$ loops and $h$ happy vertices. We are interested in establishing necessary and sufficient conditions for the class $\mathcal{G}_{n|d|h}$ to contain a labeled graph accepting the proposal, as well as conditions to contain a graph rejecting the proposal. We also discuss connections to the existing literature, including that on majority domination, and investigate the properties of the obtained conditions.Comment: 28 pages, 8 figures, accepted for publication in Discrete Applied Mathematic