20,433 research outputs found

    On global location-domination in graphs

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    A dominating set SS of a graph GG is called locating-dominating, LD-set for short, if every vertex vv not in SS is uniquely determined by the set of neighbors of vv belonging to SS. Locating-dominating sets of minimum cardinality are called LDLD-codes and the cardinality of an LD-code is the location-domination number λ(G)\lambda(G). An LD-set SS of a graph GG is global if it is an LD-set of both GG and its complement G‾\overline{G}. The global location-domination number λg(G)\lambda_g(G) is the minimum cardinality of a global LD-set of GG. In this work, we give some relations between locating-dominating sets and the location-domination number in a graph and its complement.Comment: 15 pages: 2 tables; 8 figures; 20 reference

    On global location-domination in graphs

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    A dominating set S of a graph G is called locating-dominating, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number lambda(G). An LD-set S of a graph G is global if it is an LD-set of both G and its complement G'. The global location-domination number lambda g(G) is introduced as the minimum cardinality of a global LD-set of G. In this paper, some general relations between LD-codes and the location-domination number in a graph and its complement are presented first. Next, a number of basic properties involving the global location-domination number are showed. Finally, both parameters are studied in-depth for the family of block-cactus graphs.Postprint (published version

    Solving Two Conjectures regarding Codes for Location in Circulant Graphs

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    Identifying and locating-dominating codes have been widely studied in circulant graphs of type Cn(1,2,…,r)C_n(1,2, \ldots, r), which can also be viewed as power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered identification and location-domination in the circulant graphs Cn(1,3)C_n(1,3). They showed that the smallest cardinality of a locating-dominating code in Cn(1,3)C_n(1,3) is at least ⌈n/3⌉\lceil n/3 \rceil and at most ⌈n/3⌉+1\lceil n/3 \rceil + 1 for all n≥9n \geq 9. Moreover, they proved that the lower bound is strict when n≡0,1,4(mod6)n \equiv 0, 1, 4 \pmod{6} and conjectured that the lower bound can be increased by one for other nn. In this paper, we prove their conjecture. Similarly, they showed that the smallest cardinality of an identifying code in Cn(1,3)C_n(1,3) is at least ⌈4n/11⌉\lceil 4n/11 \rceil and at most ⌈4n/11⌉+1\lceil 4n/11 \rceil + 1 for all n≥11n \geq 11. Furthermore, they proved that the lower bound is attained for most of the lengths nn and conjectured that in the rest of the cases the lower bound can improved by one. This conjecture is also proved in the paper. The proofs of the conjectures are based on a novel approach which, instead of making use of the local properties of the graphs as is usual to identification and location-domination, also manages to take advantage of the global properties of the codes and the underlying graphs

    Solving Two Conjectures regarding Codes for Location in Circulant Graphs

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    Identifying and locating-dominating codes have been widely studied in circulant graphs of type Cn(1, 2, . . ., r), which can also be viewed as power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered identification and location-domination in the circulant graphs Cn(1, 3). They showed that the smallest cardinality of a locating-dominating code in Cn(1, 3) is at least ⌈n/3⌉ and at most ⌈n/3⌉ + 1 for all n ≥ 9. Moreover, they proved that the lower bound is strict when n ≡ 0, 1, 4 (mod 6) and conjectured that the lower bound can be increased by one for other n. In this paper, we prove their conjecture. Similarly, they showed that the smallest cardinality of an identifying code in Cn(1, 3) is at least ⌈4n/11⌉ and at most ⌈4n/11⌉ + 1 for all n ≥ 11. Furthermore, they proved that the lower bound is attained for most of the lengths n and conjectured that in the rest of the cases the lower bound can improved by one. This conjecture is also proved in the paper. The proofs of the conjectures are based on a novel approach which, instead of making use of the local properties of the graphs as is usual to identification and location-domination, also manages to take advantage of the global properties of the codes and the underlying graphs.</p

    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

    Finding maximum k-cliques faster using lazy global domination

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    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
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