16,359 research outputs found

    Protecting a Graph with Mobile Guards

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    Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

    A note on the independent roman domination in unicyclic graphs

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    A Roman dominating function (RDF) on a graph G=(V;E)G = (V;E) is a function f:V{0,1,2}f : V \to \{0, 1, 2\} satisfying the condition that every vertex uu for which f(u)=0f(u) = 0 is adjacent to at least one vertex vv for which f(v)=2f(v) = 2. The weight of an RDF is the value f(V(G))=uV(G)f(u)f(V(G)) = \sum _{u \in V (G)} f(u). An RDF ff in a graph GG is independent if no two vertices assigned positive values are adjacent. The Roman domination number γR(G)\gamma _R (G) (respectively, the independent Roman domination number iR(G)i_{R}(G)) is the minimum weight of an RDF (respectively, independent RDF) on GG. We say that γR(G)\gamma _R (G) strongly equals iR(G)i_R (G), denoted by γR(G)iR(G)\gamma _R (G) \equiv i_R (G), if every RDF on GG of minimum weight is independent. In this note we characterize all unicyclic graphs GG with γR(G)iR(G)\gamma _R (G) \equiv i_R (G)

    Trees with Unique Italian Dominating Functions of Minimum Weight

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    An Italian dominating function, abbreviated IDF, of GG is a function f ⁣:V(G){0,1,2}f \colon V(G) \rightarrow \{0, 1, 2\} satisfying the condition that for every vertex vV(G)v \in V(G) with f(v)=0f(v)=0, we have uN(v)f(u)2\sum_{u \in N(v)} f(u) \ge 2. That is, either vv is adjacent to at least one vertex uu with f(u)=2f(u) = 2, or to at least two vertices xx and yy with f(x)=f(y)=1f(x) = f(y) = 1. The Italian domination number, denoted γI\gamma_I(G), is the minimum weight of an IDF in GG. In this thesis, we use operations that join two trees with a single edge in order to build trees with unique γI\gamma_I-functions

    The total co-independent domination number of some graph operations

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    [EN] A set D of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex of D. The total dominating set D is called a total co-independent dominating set if the subgraph induced by V (G)- D is edgeless. The minimum cardinality among all total co-independent dominating sets of G is the total co-independent domination number of G. In this article we study the total co-independent domination number of the join, strong, lexicographic, direct and rooted products of graphs.I. Peterin was partially supported by ARRS Slovenia under grants P1-0297 and J1-9109; I. G. Yero was partially supported by Junta de Andalucia, FEDER-UPO Research and Development Call, reference number UPO-1263769.Cabrera Martinez, A.; Cabrera García, S.; Peterin, I.; Yero, IG. (2022). The total co-independent domination number of some graph operations. Revista de la Unión Matemática Argentina. 63(1):153-158. https://doi.org/10.33044/revuma.165215315863
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