14,287 research outputs found

    Total Roman {2}-domination in graphs

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    [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 García, 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

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    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

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    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 kk-monopolies in graphs which are closely related to different parameters in graphs. Given a graph G=(V,E)G=(V,E) and X⊆VX\subseteq V, if δX(v)\delta_X(v) is the number of neighbors vv has in XX, kk is an integer and tt is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set M⊆VM\subseteq V a vertex vv of GG is said to be kk-controlled by MM if δM(v)≥δV(v)2+k\delta_M(v)\ge \frac{\delta_V(v)}{2}+k. The set MM is called an open kk-monopoly for GG if it kk-controls every vertex vv of GG. - A function f:V→{−1,1}f: V\rightarrow \{-1,1\} is called a signed total tt-dominating function for GG if f(N(v))=∑v∈N(v)f(v)≥tf(N(v))=\sum_{v\in N(v)}f(v)\geq t for all v∈Vv\in V. - A nonempty set S⊆VS\subseteq V is a global (defensive and offensive) kk-alliance in GG if δS(v)≥δV−S(v)+k\delta_S(v)\ge \delta_{V-S}(v)+k holds for every v∈Vv\in V. In this article we prove that the problem of computing the minimum cardinality of an open 00-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 kk-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

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    Let G=(V,E)G=(V,E) be a graph. A subset D⊆VD\subseteq V is a dominating set if every vertex not in DD is adjacent to a vertex in DD. The domination number of GG, denoted by γ(G)\gamma(G), is the smallest cardinality of a dominating set of GG. The bondage number of a nonempty graph GG is the smallest number of edges whose removal from GG results in a graph with domination number larger than γ(G)\gamma(G). The reinforcement number of GG is the smallest number of edges whose addition to GG results in a graph with smaller domination number than γ(G)\gamma(G). 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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