5 research outputs found
Deploying robots with two sensors in -free graphs
Let be a graph of minimum degree at least two with no induced subgraph
isomorphic to . We prove that if is not isomorphic to one of eight
exceptional graphs, then it is possible to assign two-element subsets of
to the vertices of in such a way that for every
and every vertex the label is assigned to
or one of its neighbors. It follows that has fractional domatic number
at least . This is motivated by a problem in robotics and generalizes a
result of Fujita, Yamashita and Kameda who proved that the same conclusion
holds for all -regular graphs
Common extremal graphs for three inequalities involving domination parameters
‎Let ‎, ‎ and ‎ ‎be the minimum degree‎, ‎maximum degree and‎ ‎domination number of a graph ‎, ‎respectively‎. ‎A partition of ‎, ‎all of whose classes are dominating sets in ‎, ‎is called a domatic partition of ‎. ‎The maximum number of classes of‎ ‎a domatic partition of is called the domatic number of ‎, ‎denoted ‎. ‎It is well known that‎ ‎‎, ‎ cite{ch}‎, ‎and cite{berge}‎. ‎In this paper‎, ‎we investigate the graphs for which‎ ‎all the above inequalities become simultaneously equalities‎
Characterizing Heterogeneity in Cooperative Networks From a Resource Distribution View-Point
© by International Press. First published in Communications in Information and Systems, Vol. 14, no. 1, 2014, by International Press.DOI: http://dx.doi.org/10.4310/CIS.2014.v14.n1.a1A network of agents in which agents with a diverse set of resources
or capabilities interact and coordinate with each other to accomplish various tasks constitutes a heterogeneous cooperative network. In this paper, we investigate heterogeneity in terms of resources allocated to agents within the network. The objective is to distribute resources in such a way that every agent in the network
should be able to utilize all these resources through local interactions. In particular, we formulate a graph coloring problem in which
each node is assigned a subset of labels from a labeling set, and a graph is considered to be completely heterogeneous whenever all the labels in the labeling set are available in the closed neighborhood of every node. The total number of different resources that can be accommodated within a system under this setting depends
on the underlying graph structure of the network. This paper provides an analysis of the assignment of multiple resources to nodes and the effect of these assignments on the overall heterogeneity of the network
A New Optimality Measure for Distance Dominating Sets
  We study the problem of finding the smallest power of an input graph that has k disjoint dominating sets, where the ith power of an input graph G is constructed by adding edges between pairs of vertices in G at distance i or less, and a subset of vertices in a graph G is a dominating set if and only if every vertex in G is adjacent to a vertex in this subset.
  The problem is a different view of the d-domatic number problem in which the goal is to find the maximum number of disjoint dominating sets in the dth power of the input graph.
  This problem is motivated by applications in multi-facility location and distributed networks. In the facility location framework, for instance, there are k types of services that all clients in different regions of a city should receive. A graph representing the map of regions in the city is given where the nodes of the graph represent regions and neighboring regions are connected by edges. The problem is how to establish facility servers in the city (each region can host at most one server) such that every client in the city can access a facility server in its region or in a region in the neighborhood. Since it may not be possible to find a facility location satisfying this condition, "a region in the neighborhood" required in the question is modified to "a region at the minimum possible distance d".
  In this thesis, we study the connection of the above-mentioned problem with similar problems including the domatic number problem and the d-domatic number problem. We show that the problem is NP-complete for any fixed k greater than two even when the input graph is restricted to split graphs, 2-connected graphs, or planar bipartite graphs of degree four. In addition, the problem is in P for bounded tree-width graphs, when considering k as a constant, and for strongly chordal graphs, for any k. Then, we provide a slightly simpler proof for a known upper bound for the problem. We also develop an exact (exponential) algorithm for the problem, running in time O(2. 73n). Moreover, we prove that the problem cannot be approximated within ratio smaller than 2 even for split graphs, 2-connected graphs, and planar bipartite graphs of degree four. We propose a greedy 3-approximation algorithm for the problem in the general case, and other approximation ratios for permutation graphs, distance-hereditary graphs, cocomparability graphs, dually chordal graphs, and chordal graphs. Finally, we list some directions for future work