  We investigate the impact of attractive-repulsive interaction in networks of
limit-cycle oscillators. Mainly we focus on the design principle for generating
an anti-phase state between adjacent nodes in a complex network. We establish
that a partial negative control throughout the branches of a spanning tree
inside the positively coupled limit-cycle oscillators works efficiently well in
comparison with randomly chosen negative links to establish zero frustration
(anti-phase synchronization) in bipartite graphs. Based on the emergence of
zero frustration, we develop a universal 0-$\pi$ rule to understand the
anti-phase synchronization in a bipartite graph. Further, this rule is used to
construct a non-bipartite graph for a given non-zero frustrated value. We
finally show the generality of 0-$\pi$ rule by implementing it in arbitrary
undirected non-bipartite graphs of attractive-repulsively coupled limit-cycle
oscillators and successfully calculate the non-zero frustration value which
matches with numerical data. The validation of the rule is checked through the
bifurcation analysis of small networks. Our work may unveil the underlying
mechanism of several synchronization phenomena that exist in a network of
oscillators having a mixed type of coupling.
