Italian domination on Mycielskian and Sierpinski graphs

An Italian dominating function (IDF) of a graph G is a function [Formula: see text] satisfying the condition that for every [Formula: see text] with [Formula: see text] The weight of an IDF on [Formula: see text] is the sum [Formula: see text] and Italian domination number, [Formula: see text] is the minimum weight of an IDF. In this paper, we prove that [Formula: see text] where [Formula: see text] is the Mycielskian graph of [Formula: see text]. We have also studied the impact of edge addition on Italian domination number. We also obtain a bound for the Italian domination number of Sierpinski graph [Formula: see text] and find the exact value of [Formula: see text].

The Energy Change of the Complete Multipartite Graph

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Akbari et al. [S. Akbari, E. Ghorbani, and M. Oboudi. Edge addition, singular values, and energy of graphs and matrices. {\em Linear Algebra Appl.}, 430:2192--2199, 2009.] proved that for a complete multipartite graph $K_{t_1 ,\ldots,t_k}$, if $t_i\geq 2 \ (i=1,\ldots,k)$, then deleting any edge will increase the energy. A natural question is how the energy changes when $\min\{t_1 ,\ldots,t_k\}=1$. In this paper, a new method to study the energy of graph is explored. As an application of this new method, the above natural question is answered and it is completely determined how the energy of a complete multipartite graph changes when one edge is removed.

Election Control in Social Networks via Edge Addition or Removal

We focus on the scenario in which messages pro and/or against one or multiple candidates are spread through a social network in order to affect the votes of the receivers. Several results are known in the literature when the manipulator can make seeding by buying influencers. In this paper, instead, we assume the set of influencers and their messages to be given, and we ask whether a manipulator (e.g., the platform) can alter the outcome of the election by adding or removing edges in the social network. We study a wide range of cases distinguishing for the number of candidates or for the kind of messages spread over the network. We provide a positive result, showing that, except for trivial cases, manipulation is not affordable, the optimization problem being hard even if the manipulator has an unlimited budget (i.e., he can add or remove as many edges as desired). Furthermore, we prove that our hardness results still hold in a reoptimization variant, where the manipulator already knows an optimal solution to the problem and needs to compute a new solution once a local modification occurs (e.g., in bandit scenarios where estimations related to random variables change over time).

Split Domination Vertex Critical and Edge Critical Graphs

A dominating set D of a graph G = (V;E) is a split dominating set ifthe induced graph hV 􀀀 Di is disconnected. The split domination number s(G)is the minimum cardinality of a split domination set. A graph G is called vertexsplit domination critical if s(G􀀀v) s(G) for every vertex v 2 G. A graph G iscalled edge split domination critical if s(G + e) s(G) for every edge e in G. Inthis paper, whether for some standard graphs are split domination vertex critical ornot are investigated and then characterized 2- ns-critical and 3- ns-critical graphswith respect to the diameter of a graph G with vertex removal. Further, it is shownthat there is no existence of s-critical graph for edge addition.

A Novel Graph-modification Technique for User Privacy-preserving on Social Networks

The growing popularity of social networks and the increasing need for publishing related data mean that protection of privacy becomes an important and challenging problem in social networks. This paper describes the (k,l k,l k,l)-anonymity model used for social network graph anonymization. The method is based on edge addition and is utility-aware, i.e. it is designed to generate a graph that is similar to the original one. Different strategies are evaluated to this end and the results are compared based on common utility metrics. The outputs confirm that the na¨ıve idea of adding some random or even minimum number of possible edges does not always produce useful anonymized social network graphs, thus creating some interesting alternatives for graph anonymization techniques.

K-Core Maximization: An Edge Addition Approach

A popular model to measure the stability of a network is k-core - the maximal induced subgraph in which every vertex has at least k neighbors. Many studies maximize the number of vertices in k-core to improve the stability of a network. In this paper, we study the edge k-core problem: Given a graph G, an integer k and a budget b, add b edges to non-adjacent vertex pairs in G such that the k-core is maximized. We prove the problem is NP-hard and APX-hard. A heuristic algorithm is proposed on general graphs with effective optimization techniques. Comprehensive experiments on 9 real-life datasets demonstrate the effectiveness and the efficiency of our proposed methods.