1,245 research outputs found

    Location-domination in line graphs

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    A set DD of vertices of a graph GG is locating if every two distinct vertices outside DD have distinct neighbors in DD; that is, for distinct vertices uu and vv outside DD, N(u)DN(v)DN(u) \cap D \neq N(v) \cap D, where N(u)N(u) denotes the open neighborhood of uu. If DD is also a dominating set (total dominating set), it is called a locating-dominating set (respectively, locating-total dominating set) of GG. A graph GG is twin-free if every two distinct vertices of GG have distinct open and closed neighborhoods. It is conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning. Locating-total dominating sets in twin-free graphs: a conjecture. The Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any twin-free graph GG without isolated vertices has a locating-dominating set of size at most one-half its order and a locating-total dominating set of size at most two-thirds its order. In this paper, we prove these two conjectures for the class of line graphs. Both bounds are tight for this class, in the sense that there are infinitely many connected line graphs for which equality holds in the bounds.Comment: 23 pages, 2 figure

    Stochastic domination for the Ising and fuzzy Potts models

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    We discuss various aspects concerning stochastic domination for the Ising model and the fuzzy Potts model. We begin by considering the Ising model on the homogeneous tree of degree dd, \Td. For given interaction parameters J1J_1, J2>0J_2>0 and external field h_1\in\RR, we compute the smallest external field h~\tilde{h} such that the plus measure with parameters J2J_2 and hh dominates the plus measure with parameters J1J_1 and h1h_1 for all hh~h\geq\tilde{h}. Moreover, we discuss continuity of h~\tilde{h} with respect to the three parameters J1J_1, J2J_2, hh and also how the plus measures are stochastically ordered in the interaction parameter for a fixed external field. Next, we consider the fuzzy Potts model and prove that on \Zd the fuzzy Potts measures dominate the same set of product measures while on \Td, for certain parameter values, the free and minus fuzzy Potts measures dominate different product measures. For the Ising model, Liggett and Steif proved that on \Zd the plus measures dominate the same set of product measures while on \T^2 that statement fails completely except when there is a unique phase.Comment: 22 pages, 5 figure
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