14,287 research outputs found
Total Roman {2}-domination in graphs
[EN] Given a graph G = (V, E), a function f: V -> {0, 1, 2} is a total Roman {2}-dominating function if every vertex v is an element of V for which f (v) = 0 satisfies that n-ary sumation (u)(is an element of N (v)) f (v) >= 2, where N (v) represents the open neighborhood of v, and every vertex x is an element of V for which f (x) >= 1 is adjacent to at least one vertex y is an element of V such that f (y) >= 1. The weight of the function f is defined as omega(f ) = n-ary sumation (v)(is an element of V) f (v). The total Roman {2}-domination number, denoted by gamma(t)({R2})(G), is the minimum weight among all total Roman {2}-dominating functions on G. In this article we introduce the concepts above and begin the study of its combinatorial and computational properties. For instance, we give several closed relationships between this parameter and other domination related parameters in graphs. In addition, we prove that the complexity of computing the value gamma(t)({R2})(G) is NP-hard, even when restricted to bipartite or chordal graphsCabrera GarcÃa, S.; Cabrera Martinez, A.; Hernandez Mira, FA.; Yero, IG. (2021). Total Roman {2}-domination in graphs. Quaestiones Mathematicae. 44(3):411-444. https://doi.org/10.2989/16073606.2019.1695230S41144444
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Open k-monopolies in graphs: complexity and related concepts
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 -monopolies in graphs which are closely related to different parameters
in graphs. Given a graph and , if is the
number of neighbors has in , is an integer and is a positive
integer, then we establish in this article a connection between the following
three concepts:
- Given a nonempty set a vertex of is said to be
-controlled by if . The set
is called an open -monopoly for if it -controls every vertex of
.
- A function is called a signed total
-dominating function for if for all
.
- A nonempty set is a global (defensive and offensive)
-alliance in if holds for every .
In this article we prove that the problem of computing the minimum
cardinality of an open -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 -monopolies and we derive some exact values.Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016
The Algorithmic Complexity of Bondage and Reinforcement Problems in bipartite graphs
Let be a graph. A subset is a dominating set if
every vertex not in is adjacent to a vertex in . The domination number
of , denoted by , is the smallest cardinality of a dominating set
of . The bondage number of a nonempty graph is the smallest number of
edges whose removal from results in a graph with domination number larger
than . The reinforcement number of is the smallest number of
edges whose addition to results in a graph with smaller domination number
than . In 2012, Hu and Xu proved that the decision problems for the
bondage, the total bondage, the reinforcement and the total reinforcement
numbers are all NP-hard in general graphs. In this paper, we improve these
results to bipartite graphs.Comment: 13 pages, 4 figures. arXiv admin note: substantial text overlap with
arXiv:1109.1657; and text overlap with arXiv:1204.4010 by other author
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