We study the vulnerability of dominating sets against random and targeted node removals in complex networks. While small, cost-efficient dominating sets play a significant role in controllability and observability of these networks, a fixed and intact network structure is always implicitly assumed. We find that cost-efficiency of dominating sets optimized for small size alone comes at a price o...

A {em Roman dominating function} on a graph $G$ is a function$f:V(G)rightarrow {0,1,2}$ satisfying the condition that everyvertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex$v$ for which $f(v) =2$. {color{blue}A {em restrained Roman dominating}function} $f$ is a {color{blue} Roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} The wei...

The great puzzle of Caribbean history in the mid-19th century is that this was a time of radical institutional change – with all but one of the long-established Caribbean parliaments voting themselves out of existence inside a twenty-year window – despite very stable conditions in the (exogenous) terms-of-trade and in the (endogenous) local plantation economy. This paper argues that the post-sl...

Summary The politeia established in Athens 317 after a covenant between Cassander and Demetrius of Phalerum was variously described by ancient authors as tyranny, an oligarchy, democracy. Even among modern scholars there is no agreement about its definition. A close study the architecture so-called regime comparison with almost contemporary constitution Cyrene, imposed on African town Ptolemaeu...

The phenomenon of a single candidate lately happens lot in Indonesia, especially the sphere local election democratic parties. In elections held simultaneously Indonesia 2017, many regions were only attended by one pair. This makes climate competition democracy not as expected with democracy. Democracy aims to make elite circulation happen, so that ruling does dominate it becomes an oligarchy. ...

A subset $S$ of vertex set $V(D)$ is an {em indpendent dominating set} of $D$ if $S$ is both an independent and a dominating set of $D$. The {em indpendent domination number}, $i(D)$ is the cardinality of the smallest independent dominating set of $D$. In this paper we calculate the independent domination number of the { em cartesian product} of two {em directed paths} $P_m$ and $P_n$ for arbi...

A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...