8,323 research outputs found

    Bounds for identifying codes in terms of degree parameters

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    An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If \M(G) denotes the minimum size of an identifying code of a graph GG, it was conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there exists a constant cc such that if a connected graph GG with nn vertices and maximum degree dd admits an identifying code, then \M(G)\leq n-\tfrac{n}{d}+c. We use probabilistic tools to show that for any d3d\geq 3, \M(G)\leq n-\tfrac{n}{\Theta(d)} holds for a large class of graphs containing, among others, all regular graphs and all graphs of bounded clique number. This settles the conjecture (up to constants) for these classes of graphs. In the general case, we prove \M(G)\leq n-\tfrac{n}{\Theta(d^{3})}. In a second part, we prove that in any graph GG of minimum degree δ\delta and girth at least 5, \M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n. Using the former result, we give sharp estimates for the size of the minimum identifying code of random dd-regular graphs, which is about logddn\tfrac{\log d}{d}n

    Vertex identifying codes for the n-dimensional lattice

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    An rr-identifying code on a graph GG is a set CV(G)C\subset V(G) such that for every vertex in V(G)V(G), the intersection of the radius-rr closed neighborhood with CC is nonempty and different. Here, we provide an overview on codes for the nn-dimensional lattice, discussing the case of 1-identifying codes, constructing a sparse code for the 4-dimensional lattice as well as showing that for fixed nn, the minimum density of an rr-identifying code is Θ(1/rn1)\Theta(1/r^{n-1}).Comment: 10p

    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

    Improved Bounds for rr-Identifying Codes of the Hex Grid

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    For any positive integer rr, an rr-identifying code on a graph GG is a set CV(G)C\subset V(G) such that for every vertex in V(G)V(G), the intersection of the radius-rr closed neighborhood with CC is nonempty and pairwise distinct. For a finite graph, the density of a code is C/V(G)|C|/|V(G)|, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We find a code of density less than 5/(6r)5/(6r), which is sparser than the prior best construction which has density approximately 8/(9r)8/(9r).Comment: 12p
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