2,162 research outputs found
Countable locally 2-arc-transitive bipartite graphs
We present an order-theoretic approach to the study of countably infinite
locally 2-arc-transitive bipartite graphs. Our approach is motivated by
techniques developed by Warren and others during the study of cycle-free
partial orders. We give several new families of previously unknown countably
infinite locally-2-arc-transitive graphs, each family containing continuum many
members. These examples are obtained by gluing together copies of incidence
graphs of semilinear spaces, satisfying a certain symmetry property, in a
tree-like way. In one case we show how the classification problem for that
family relates to the problem of determining a certain family of highly
arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page
Global offensive -alliances in digraphs
In this paper, we initiate the study of global offensive -alliances in
digraphs. Given a digraph , a global offensive -alliance in a
digraph is a subset such that every vertex outside of
has at least one in-neighbor from and also at least more in-neighbors
from than from outside of , by assuming is an integer lying between
two minus the maximum in-degree of and the maximum in-degree of . The
global offensive -alliance number is the minimum
cardinality among all global offensive -alliances in . In this article we
begin the study of the global offensive -alliance number of digraphs. For
instance, we prove that finding the global offensive -alliance number of
digraphs is an NP-hard problem for any value and that it remains NP-complete even when
restricted to bipartite digraphs when we consider the non-negative values of
given in the interval above. Based on these facts, lower bounds on
with characterizations of all digraphs attaining the bounds
are given in this work. We also bound this parameter for bipartite digraphs
from above. For the particular case , an immediate result from the
definition shows that for all digraphs ,
in which stands for the domination number of . We show that
these two digraph parameters are the same for some infinite families of
digraphs like rooted trees and contrafunctional digraphs. Moreover, we show
that the difference between and can be
arbitrary large for directed trees and connected functional digraphs
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