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Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article, we study the complexity of #(p) HOMSTOH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p. Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of H on the tractability was shown to persist, which yields similar dichotomies. <br /> Our main result states that for every tree H and every prime p the problem #pHOMSTOH is either polynomial time computable or #P-p-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of #pHOMSTOH are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime p. These results are the first suggesting that such dichotomies hold not only for the modulo 2 case but also for the modular counting functions of all primes p.
Criteria of talent in sport
(1999)
Einführung in den Band
(2019)
Similar to maternal care, paternal care is a source of neonatal sensory stimulation, which in primates and rodents has been shown to be essential for developing structure and function of sensory cortices. The aim of our study in the biparental rodent Octodon degus was to assess the impact of paternal deprivation on dendritic and synaptic development in the somatosensory cortex. We (i) quantified the amount of paternal care in relation to total parental investment and (ii) compared dendritic and synaptic development of pyramidal neurons in the somatosensory cortex of animals raised by a single mother or by both parents. On the behavioral level we show that paternal care comprises 37% of total parent-offspring interactions, and that the somatosensory stimulation provided by the fathers primarily consists of huddling, licking/grooming, and playing. On the morphological level we found that, compared with offspring raised by both parents (mother and father), the father-deprived animals displayed significantly reduced spine numbers on the basal dendrites of pyramidal neurons. Furthermore, paternal deprivation induces hemispheric asymmetry of the dendritic morphology of somatosensory pyramidal neurons. Father-deprived animals show shorter and less complex basal dendrites in the left somatosensory cortex compared with the right hemisphere. These findings indicate that paternal deprivation results in delayed or retarded dendritic and synaptic development of somatosensory circuits.
Studieneingang optimieren!
(2019)