45,104 research outputs found

    Identifying codes of corona product graphs

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    For a vertex xx of a graph GG, let NG[x]N_G[x] be the set of xx with all of its neighbors in GG. A set CC of vertices is an {\em identifying code} of GG if the sets NG[x]∩CN_G[x]\cap C are nonempty and distinct for all vertices xx. If GG admits an identifying code, we say that GG is identifiable and denote by γID(G)\gamma^{ID}(G) the minimum cardinality of an identifying code of GG. In this paper, we study the identifying code of the corona product H⊙GH\odot G of graphs HH and GG. We first give a necessary and sufficient condition for the identifiable corona product H⊙GH\odot G, and then express γID(H⊙G)\gamma^{ID}(H\odot G) in terms of γID(G)\gamma^{ID}(G) and the (total) domination number of HH. Finally, we compute γID(H⊙G)\gamma^{ID}(H\odot G) for some special graphs GG

    Identifying codes in vertex-transitive graphs and strongly regular graphs

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    We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs

    Partitioning the vertex set of GG to make G □ HG\,\Box\, H an efficient open domination graph

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    A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs GG for which the Cartesian product Gâ–¡HG \Box H is an efficient open domination graph when HH is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of V(G)V(G). For the class of trees when HH is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products Gâ–¡HG \Box H when HH is a 5-cycle or a 4-cycle.Comment: 16 pages, 2 figure

    Loop Calculus in Statistical Physics and Information Science

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    Considering a discrete and finite statistical model of a general position we introduce an exact expression for the partition function in terms of a finite series. The leading term in the series is the Bethe-Peierls (Belief Propagation)-BP contribution, the rest are expressed as loop-contributions on the factor graph and calculated directly using the BP solution. The series unveils a small parameter that often makes the BP approximation so successful. Applications of the loop calculus in statistical physics and information science are discussed.Comment: 4 pages, submitted to Phys.Rev.Lett. Changes: More general model, Simpler derivatio

    Homological Error Correction: Classical and Quantum Codes

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    We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension D>2D>2, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension DD. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
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