Dorota Kuziak ; Iztok Peterin ; Ismael G. Yero - Open k-monopolies in graphs: complexity and related concepts

dmtcs:654 - Discrete Mathematics & Theoretical Computer Science, March 29, 2016, Vol. 18 no. 3 -
Open k-monopolies in graphs: complexity and related conceptsArticle

Authors: Dorota Kuziak ORCID; Iztok Peterin ORCID; 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.

    Volume: Vol. 18 no. 3
    Section: Graph Theory
    Published on: March 29, 2016
    Submitted on: March 10, 2016
    Keywords: Mathematics - Combinatorics,05C85, 05C69, 05C07

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