2 research outputs found
Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees
Let be a graph, and let . For a set of vertices of
, let the set arise by starting with the set , and
iteratively adding further vertices to the current set if they have at
least neighbors in it. If contains all
vertices of , then is known as an irreversible dynamic monopoly or a
perfect target set associated with the threshold function . Let be the minimum cardinality of such an
irreversible dynamic monopoly.
For a connected graph of maximum degree at least , Chang
(Triggering cascades on undirected connected graphs, Information Processing
Letters 111 (2011) 973-978) showed , which was
improved by Chang and Lyuu (Triggering cascades on strongly connected directed
graphs, Theoretical Computer Science 593 (2015) 62-69) to . We show that for every , there is some
such that for every
in , and every connected graph that has maximum
degree at least and girth at least . Furthermore, we show
that for every in , and every tree
that has order at least
Vaccinate your trees!
For a graph and an integer-valued function on its vertex set, a
dynamic monopoly is a set of vertices of such that iteratively adding to it
vertices of that have at least neighbors in it eventually
yields the vertex set of . We study two vaccination problems, where the goal
is to maximize the minimum order of such a dynamic monopoly either by
increasing the threshold value of vertices beyond their degree, or by
removing vertices from , where is a given non-negative integer
corresponding to a budget. We show how to solve these problems efficiently for
trees