Open k-monopolies in graphs: complexity and related concepts
Authors: Dorota Kuziak ; Iztok Peterin ; Ismael G. Yero
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Dorota Kuziak;Iztok Peterin;Ismael G. Yero
Closed monopolies in graphs have a quite long range of applications in
several problems related to overcoming failures, since they frequently have
some common approaches around the notion of majorities, for instance to
consensus problems, diagnosis problems or voting systems. We introduce here
open $k$-monopolies in graphs which are closely related to different parameters
in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $\delta_X(v)$ is the
number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive
integer, then we establish in this article a connection between the following
three concepts:
- Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be
$k$-controlled by $M$ if $\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$. The set $M$
is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of
$G$.
- A function $f: V\rightarrow \{-1,1\}$ is called a signed total
$t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all
$v\in V$.
- A nonempty set $S\subseteq V$ is a global (defensive and offensive)
$k$-alliance in $G$ if $\delta_S(v)\ge \delta_{V-S}(v)+k$ holds for every $v\in
V$.
In this article we prove that the problem of computing the minimum
cardinality of an open $0$-monopoly in a graph is NP-complete even restricted
to bipartite or chordal graphs. In addition we present some general bounds for
the minimum cardinality of open $k$-monopolies and we derive some exact values.