Bounding signed series-parallel graphs and cores of signed \(K_4\)-subdivisions

Abstract

We study homomorphism properties of signed \( K_4 \)-minor-free graphs. On the one hand, we give a necessary and sufficient condition for a signed graph \( B \) to admit a homomorphism from any signed \( K_4 \)-minor-free graph and we determine the smallest of all such signed graphs. On the other hand, we characterize the minimal cores that do not belong to the class of signed \( K_4 \)-minor-free graphs. This, in particular, gives a classification of odd-\( K_4 \)’s that are cores. Furthermore, we show some applications of our work.