3,512 research outputs found

    Random subgraphs make identification affordable

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    An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with respect to graph inclusion. We show that every graph GG with nn vertices, maximum degree Δ=ω(1)\Delta=\omega(1) and minimum degree δclogΔ\delta\geq c\log{\Delta}, for some constant c>0c>0, contains a large spanning subgraph which admits an identifying code with size O(nlogΔδ)O\left(\frac{n\log{\Delta}}{\delta}\right). In particular, if δ=Θ(n)\delta=\Theta(n), then GG has a dense spanning subgraph with identifying code O(logn)O\left(\log n\right), namely, of asymptotically optimal size. The subgraph we build is created using a probabilistic approach, and we use an interplay of various random methods to analyze it. Moreover we show that the result is essentially best possible, both in terms of the number of deleted edges and the size of the identifying code

    On bounding the difference between the maximum degree and the chromatic number by a constant

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    We provide a finite forbidden induced subgraph characterization for the graph class Υk\varUpsilon_k, for all kN0k \in \mathbb{N}_0, which is defined as follows. A graph is in Υk\varUpsilon_k if for any induced subgraph, Δχ1+k\Delta \leq \chi -1 + k holds, where Δ\Delta is the maximum degree and χ\chi is the chromatic number of the subgraph. We compare these results with those given in [O. Schaudt, V. Weil, On bounding the difference between the maximum degree and the clique number, Graphs and Combinatorics 31(5), 1689-1702 (2015). DOI: 10.1007/s00373-014-1468-3], where we studied the graph class Ωk\varOmega_k, for kN0k \in \mathbb{N}_0, whose graphs are such that for any induced subgraph, Δω1+k\Delta \leq \omega -1 + k holds, where ω\omega denotes the clique number of a graph. In particular, we give a characterization in terms of Ωk\varOmega_k and Υk\varUpsilon_k of those graphs where the neighborhood of every vertex is perfect.Comment: 10 pages, 4 figure

    Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

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    A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur
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