33,518 research outputs found
A Characterization of Connected (1,2)-Domination Graphs of Tournaments
Recently. Hedetniemi et aI. introduced (1,2)-domination in graphs, and the authors extended that concept to (1, 2)-domination graphs of digraphs. Given vertices x and y in a digraph D, x and y form a (1,2)-dominating pair if and only if for every other vertex z in D, z is one step away from x or y and at most two steps away from the other. The (1,2)-dominating graph of D, dom1,2 (D), is defined to be the graph G = (V, E ) , where V (G) = V (D), and xy is an edge of G whenever x and y form a (1,2)-dominating pair in D. In this paper, we characterize all connected graphs that can be (I, 2)-dominating graphs of tournaments
On Error-detecting Open-locating-dominating sets
An open-dominating set S for a graph G is a subset of vertices where every
vertex has a neighbor in S. An open-locating-dominating set S for a graph G is
an open-dominating set such that each pair of distinct vertices in G have
distinct set of open-neighbors in S. We consider a type of a fault-tolerant
open-locating dominating set called error-detecting open-locating-dominating
sets. We present more results on the topic including its NP-completeness proof,
extremal graphs, and a characterization of cubic graphs that permit an
error-detecting open-locating-dominating set
Reconfiguration of Dominating Sets
We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph is a set of vertices such that each vertex is
either in or has a neighbour in . In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions and such that each pair of
consecutive solutions is adjacent according to a specified adjacency relation.
Two dominating sets are adjacent if one can be formed from the other by the
addition or deletion of a single vertex.
For various values of , we consider properties of , the graph
consisting of a vertex for each dominating set of size at most and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that is not necessarily
connected, for the maximum cardinality of a minimal dominating set
in . The result holds even when graphs are constrained to be planar, of
bounded tree-width, or -partite for . Moreover, we construct an
infinite family of graphs such that has exponential
diameter, for the minimum size of a dominating set. On the positive
side, we show that is connected and of linear diameter for any
graph on vertices having at least independent edges.Comment: 12 pages, 4 figure
Local Approximation Schemes for Ad Hoc and Sensor Networks
We present two local approaches that yield polynomial-time approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1+ε)-approximation to the problems at hand for any given ε > 0. The time complexity of both algorithms is O(TMIS + log*! n/εO(1)), where TMIS is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pair-wise independent nodes in every r-neighborhood is at most polynomial in r. Such graphs of polynomially bounded growth are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs
Locating domination in bipartite graphs and their complements
A set S of vertices of a graph G is distinguishing if the sets of neighbors in S for every pair of vertices not in S are distinct. A locating-dominating set of G is a dominating distinguishing set. The location-domination number of G , ¿ ( G ), is the minimum cardinality of a locating-dominating set. In this work we study relationships between ¿ ( G ) and ¿ ( G ) for bipartite graphs. The main result is the characterization of all connected bipartite graphs G satisfying ¿ ( G ) = ¿ ( G ) + 1. To this aim, we define an edge-labeled graph G S associated with a distinguishing set S that turns out to be very helpfulPostprint (author's final draft
Parameterized and approximation complexity of the detection pair problem in graphs
We study the complexity of the problem DETECTION PAIR. A detection pair of a
graph is a pair of sets of detectors with , the
watchers, and , the listeners, such that for every pair
of vertices that are not dominated by a watcher of , there is a listener of
whose distances to and to are different. The goal is to minimize
. This problem generalizes the two classic problems DOMINATING SET and
METRIC DIMENSION, that correspond to the restrictions and
, respectively. DETECTION PAIR was recently introduced by Finbow,
Hartnell and Young [A. S. Finbow, B. L. Hartnell and J. R. Young. The
complexity of monitoring a network with both watchers and listeners.
Manuscript, 2015], who proved it to be NP-complete on trees, a surprising
result given that both DOMINATING SET and METRIC DIMENSION are known to be
linear-time solvable on trees. It follows from an existing reduction by Hartung
and Nichterlein for METRIC DIMENSION that even on bipartite subcubic graphs of
arbitrarily large girth, DETECTION PAIR is NP-hard to approximate within a
sub-logarithmic factor and W[2]-hard (when parameterized by solution size). We
show, using a reduction to SET COVER, that DETECTION PAIR is approximable
within a factor logarithmic in the number of vertices of the input graph. Our
two main results are a linear-time -approximation algorithm and an FPT
algorithm for DETECTION PAIR on trees.Comment: 13 page
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