Bipartite graphs with close domination and k-domination numbers

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number $\gamma_k(G)$ is the minimum cardinality of a $k$-dominating set in $G$. For any graph $G$, we know that $\gamma_k(G) \geq \gamma(G)+k-2$ where $\Delta(G)\geq k\geq 2$ and this bound is sharp for every $k\geq 2$. In this paper, we characterize bipartite graphs satisfying the equality for $k\geq 3$ and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when $k=3$. We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general

Social Networks in Determining Employment and Wages: Patterns, Dynamics, and Inequality

We develop a model where agents obtain information about job opportunities through an explicitly modeled network of social contacts. We show that an improvement in the employment status of either an agent's direct or indirect contacts leads to an increase in the agent's employment probability and expected wages, in the sense of first order stochastic dominance. A similar effect results from an increase in the network contacts of an agent. In terms of dynamics and patterns, we show that employment is positively correlated across time and agents, and the same is true for wages. Moreover, unemployment exhibits persistence in the sense of duration dependence: the probability of obtaining a job decreases in the length of time that an agent has been unemployed. Finally, we examine inequality between two groups. If staying in the labor market is costly (in opportunity costs, education costs, or skills maintenance) and one group starts with a worse employment status or a smaller network, then that group's drop-out rate will be higher and their employment prospects and wages will be persistently below that of the other group.Networks, Labor Markets, Employment, Unemployment, Wages, Wage Inequality, Drop-Out Rates, Duration Dependence

Upper bounds for domination related parameters in graphs on surfaces

AbstractIn this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum degree

Finite functorial semi-norms and representability

Functorial semi-norms are semi-normed refinements of functors such as singular (co)homology. We investigate how different types of representability affect the (non-)triviality of finite functorial semi-norms on certain functors or classes. In particular, we consider representable functors, generalised cohomology theories, and so-called weakly flexible homology classes in singular homology and l1-homology.Comment: 18 pages; v3: small changes as suggested by the referee; v2: clarified Example 4.3, added referenc

A refinement of Betti numbers in the presence of a continuous function. ( I )

We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR in the presence of a continuous real valued function. The refinement of Betti numbers consists of finite configurations of points with multiplicities in the complex plane whose total cardinality are the Betti numbers and the refinement of homology consists of configurations of vector spaces indexed by points in complex plane, with the same support as the first, whose direct sum is isomorphic to the homology. When the homology is equipped with a scalar product these vector spaces are canonically realized as mutually orthogonal subspaces of the homology. The assignments above are in analogy with the collections of eigenvalues and generalized eigenspaces of a linear map in a finite dimensional complex vector space. A number of remarkable properties of the above configurations are discussed.Comment: 24 page