In this paper we construct the graphs C(*Y* ) and D(*Y* ) with vertex sets Acyc(*Y* ) and whose connected components encode the equivalence classes. The number of connected components of these graphs are denoted κ(*Y* ) and δ(*Y* ), respectively. We characterize the structure of C(*Y* ) and D(*Y* ), show how δ(*Y* ) can be derived from κ(*Y* ), and give enumeration results for κ(*Y* ). Moreover, we show how to associate a poset structure to each κ-equivalence class, and we characterize these posets. This allows us to create a bijection from Acyc(*Y* )/∼κ to Acyc(*Y*′)/∼κ∪Acyc(*Y′′*)/∼κ, where *Y ′* and *Y ′′*denote edge deletion and edge contraction for a cycle-edge in *Y* , respectively, which in turn shows that κ(*Y* ) may be obtained by an evaluation of the Tutte polynomial at (1, 0).

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]]>Dogs in the United States are hosts to a diverse range of ticks and tick-borne pathogens, including *A. phagocytophilum*, an important emerging canine and human pathogen. Previously, a Companion Animal Parasite Council (CAPC)-sponsored workshop proposed factors purported to be associated with the infection risk for tick-transmitted pathogens in dogs in the United States, including climate conditions, socioeconomic characteristics, local topography, and vector distribution.

Approximately four million test results from routine veterinary diagnostic tests from 2011–2013, which were collected on a county level across the contiguous United States, are statistically analyzed with the proposed factors *via* logistic regression and generalized estimating equations. Spatial prevalence maps of baseline *Anaplasma* spp. prevalence are constructed from Kriging and head-banging smoothing methods.

All of the examined factors, with the exception of surface water coverage, were significantly associated with *Anaplasma* spp. prevalence. Overall, *Anaplasma* spp. prevalence increases with increasing precipitation and forestation coverage and decreases with increasing temperature, population density, relative humidity, and elevation. Interestingly, socioeconomic status and deer/vehicle collisions were positively and negatively correlated with canine *Anaplasma* seroprevalence, respectively. A spatial map of the canine *Anaplasma* hazard is an auxiliary product of the analysis. *Anaplasma* spp. prevalence is highest in New England and the Upper Midwest.

The results from the two posited statistical models (one that contains an endemic areas assumption and one that does not) are in general agreement, with the major difference being that the endemic areas model estimates a larger prevalence in Western Texas, New Mexico, and Colorado. As *A. phagocytophilum* is zoonotic, the results of this analysis could also help predict areas of high risk for human exposure to this pathogen.