Disjunctive Total Domination in Graphs

Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $\gamma_t(G)$. A set $S$ of vertices in $G$ is a disjunctive total dominating set of $G$ if every vertex is adjacent to a vertex of $S$ or has at least two vertices in $S$ at distance2 from it. The disjunctive total domination number, $\gamma^d_t(G)$, is the minimum cardinality of such a set. We observe that $\gamma^d_t(G) \le \gamma_t(G)$. We prove that if $G$ is a connected graph of order$n \ge 8$, then $\gamma^d_t(G) \le 2(n-1)/3$ and we characterize the extremal graphs. It is known that if $G$ is a connected claw-free graph of order$n$, then $\gamma_t(G) \le 2n/3$ and this upper bound is tight for arbitrarily large$n$. We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if $G$ is a connected claw-free graph of order$n > 10$, then $\gamma^d_t(G) \le 4n/7$ and we characterize the graphs achieving equality in this bound.Comment: 23 page

Algorithmic aspects of disjunctive domination in graphs

For a graph $G=(V,E)$, a set $D\subseteq V$ is called a \emph{disjunctive dominating set} of $G$ if for every vertex $v\in V\setminus D$, $v$ is either adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it. The cardinality of a minimum disjunctive dominating set of $G$ is called the \emph{disjunctive domination number} of graph $G$, and is denoted by $\gamma_{2}^{d}(G)$. The \textsc{Minimum Disjunctive Domination Problem} (MDDP) is to find a disjunctive dominating set of cardinality $\gamma_{2}^{d}(G)$. Given a positive integer $k$ and a graph $G$, the \textsc{Disjunctive Domination Decision Problem} (DDDP) is to decide whether $G$ has a disjunctive dominating set of cardinality at most $k$. In this article, we first propose a linear time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a $(\ln(\Delta^{2}+\Delta+2)+1)$-approximation algorithm for MDDP in general graphs and prove that MDDP can not be approximated within $(1-\epsilon) \ln(|V|)$ for any $\epsilon>0$ unless NP $\subseteq$ DTIME$(|V|^{O(\log \log |V|)})$. Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree $3$

Strong Dependencies between Software Components

Component-based systems often describe context requirements in terms of explicit inter-component dependencies. Studying large instances of such systems?such as free and open source software (FOSS) distributions?in terms of declared dependencies between packages is appealing. It is however also misleading when the language to express dependencies is as expressive as boolean formulae, which is often the case. In such settings, a more appropriate notion of component dependency exists: strong dependency. This paper introduces such notion as a first step towards modeling semantic, rather then syntactic, inter-component relationships. Furthermore, a notion of component sensitivity is derived from strong dependencies, with ap- plications to quality assurance and to the evaluation of upgrade risks. An empirical study of strong dependencies and sensitivity is presented, in the context of one of the largest, freely available, component-based system

Games with a Partial Permission Structure and Their Applications

This thesis contains an introduction and six chapters. The introduction helps readers to position this thesis in game theory and graph theory, and Chapter 2 presents preliminaries, including cooperative games, digraphs, and games with a permission structure. Chapters 3, 4 and 5 provide new permission approaches, and characterize some Shapley value type solutions for cooperative games under these permission approaches. More specific, Chapter 3 combines disjunctive permission and local permission features to construct the local disjunctive permission approach. This approach requires that a player needs to get permission from at least one of its direct authorizers to cooperate, which can be used to model the approval right. Chapter 4 generalizes the local disjunctive and the local conjunctive permission approaches. This generalized approach requires that a player needs to get permission from a certain number of its direct authorizers to cooperate, which is suitable to describe certain voting situations. Chapter 5 provides a generalization of the local and the global disjunctive permission approaches, which requires that a non-top player needs permission from a sequence of authorizers that include a top player, or a sequence of at least $\xi$ authorizers to cooperate. This level generalization is commonly used in management. Besides, in those chapters, we provide axiomatizations for Shapley value type solutions for cooperative games under these permission approaches. Chapters 6 and 7 focus on applications of the content studied in the previous chapters. Chapter 6 provides a template for constructing measures based on permission values. In this chapter, we first propose the disjunctive measure to estimate dominance in a digraph by applying the local disjunctive permission value to additive games with a permission structure. Next, we generalize the disjunctive measure based on weak fairness, and extend this measure to weighted digraphs (including weights on nodes and weights on arcs). Finally, we axiomatize all these measures, and apply them to some classical networks in the literature, illustrating how they can be used to identify the key nodes in digraphs. Chapter 7 models highway toll allocation problems and provides a new platform for the application of games with a permission structure. In Chapter 7, we provide three toll allocation methods based on different toll charging rules. Besides, we characterize these methods and investigate the relationships between these methods and (disjunctive and conjunctive) permission values