A signed graph can be described as an undirected graph whose edges are assigned a + or − sign; the edges are said to be positive or negative according to whether they have + or − sign, respectively.

Signed graphs have been introduced by Harary to develop and formalize a psychological theory proposed by Heider for analysing the network of relations in a group of people. In particular, balanced signed
graphs are privileged worlds where stability reigns, thanks to two disjoint subgraphs each one internally positive, while externally negative. Balance is actually hard to find in ordinary life; several relaxations are studied in
the literature in order to better suit reality. In this spirit, we introduce two novel discrete structures: fruitful coverings and antagonism colorings.

Based on a joint work with Andrea Vietri and Manuela Montangero.

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