1,501 research outputs found
Domination and location in twin-free digraphs
A dominating set in a digraph is a set of vertices such that every vertex
is either in or has an in-neighbour in . A dominating set of a
digraph is locating-dominating if every vertex not in has a unique set of
in-neighbours within . The location-domination number of a
digraph is the smallest size of a locating-dominating set of . We
investigate upper bounds on in terms of the order of . We
characterize those digraphs with location-domination number equal to the order
or the order minus one. Such digraphs always have many twins: vertices with the
same (open or closed) in-neighbourhoods. Thus, we investigate the value of
in the absence of twins and give a general method for
constructing small locating-dominating sets by the means of special dominating
sets. In this way, we show that for every twin-free digraph of order ,
holds, and there exist twin-free digraphs
with . If moreover is a tournament or is
acyclic, the bound is improved to ,
which is tight in both cases
Distances and Domination in Graphs
This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present
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