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

Decidability of Code Properties

We explore the borderline between decidability and undecidability of the following question: "Let C be a class of codes. Given a machine M of type X , is it decidable whether the language L(M) lies in C or not?" for codes in general, w-codes, codes of finite and bounded deciphering delay, prefix, suffix and bi(pre)fix codes, and for finite automata equipped with different versions of push-down stores and counters. E-mail: ffernau,[email protected] + E-mail: [email protected] 2 H. Fernau, K. Reinhardt, L. Staiger Contents 1 Introduction 3 2 Definitions 3 2.1 Counter machines . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 C-Chains 7 4 Decidability results 10 5 Undecidability results 11 5.1 Blind Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Other Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6 Summary and Prospect..