Polynomial reconstruction of signed graphs

Abstract

The reconstruction problem of the characteristic polynomial of graphs from their polynomial decks was posed in 1973. So far this problem is not resolved except for some particular cases. Moreover, no counterexample for graphs of order n>2 is known. Here we put forward the analogous problem for signed graphs, and besides some general results, we resolve it within signed trees and unicyclic signed graphs, and also within disconnected signed graphs whose one component is either a signed tree or is unicyclic. A family of counterexamples that was encountered in this paper consists of two signed cycles of the same order, one balanced and the other unbalanced.