2 research outputs found
Distance domination, guarding and vertex cover for maximal outerplanar graph
This paper discusses a distance guarding concept on triangulation graphs,
which can be associated with distance domination and distance vertex cover. We
show how these subjects are interconnected and provide tight bounds for any
n-vertex maximal outerplanar graph: the 2d-guarding number, g_{2d}(n) = n/5;
the 2d-distance domination number, gamma_{2d}(n) = n/5; and the 2d-distance
vertex cover number, beta_{2d}(n) = n/4
Dominating Sets inducing Large Components in Maximal Outerplanar Graphs
For a maximal outerplanar graph of order at least , Matheson and
Tarjan showed that has domination number at most . Similarly, for a
maximal outerplanar graph of order at least , Dorfling, Hattingh,
and Jonck showed, by a completely different approach, that has total
domination number at most unless is isomorphic to one of two
exceptional graphs of order .
We present a unified proof of a common generalization of these two results.
For every positive integer , we specify a set of graphs of
order at least and at most such that every maximal outerplanar
graph of order at least that does not belong to has
a dominating set of order at most such that
every component of the subgraph of induced by has order at least